Matrices and determinants

2.1, 2.1.1 and 2.1.2Matrices

A matrix is an array of numbers arranged in horizontal and vertical lines and enclosed within square brackets. Matrices are usually denoted by capital letters. The horizontal lines are the rows of the matrix and the vertical lines are its columns.

Each number in the matrix is called an element or entry. Every element has a definite position, given by the number of the row first and then the number of the column in which it lies. In the matrix above, the element $8$ is where the second row and the third column meet.

In general, the element in the $i$th row and the $j$th column is written $a_{ij}$, and the matrix $A$ is written $A=[a_{ij}]$. For example, in

the element $1$ is in the first row and first column ($c_{11}=1$), and $4$ is in the first row and second column ($c_{12}=4$), and so on.

Order of a matrix (2.1.1)

The order of a matrix tells how many rows and columns it has. If a matrix $A$ has $m$ rows and $n$ columns, its order is $m\times n$ (read "$m$-by-$n$"). We always write the number of rows first and the number of columns second.

Multiplying $m$ by $n$ gives the total number of elements. For example, the matrix $A$ above has 2 rows and 3 columns, so its order is $2\times 3$, often written $A_{2\times 3}$, and it has $2\times 3=6$ elements.

Equality of matrices (2.1.2)

Two matrices are equal if and only if both have the same order and the same corresponding elements.

Take

Both are of order $3\times 2$ and have the same corresponding elements, so $A=B$. Equality is about the values of the entries, not the way they are written. Hence

2.1.3 to 2.1.14Types of matricesOften examined

Matrices are named by the shape or by the pattern of their entries. The book describes the following types. For each type, learn the definition and one small example.

Row matrix or row vector (2.1.3)

A matrix with only one row is a row matrix or row vector. Examples:

In general, a row matrix with $n$ columns is written

Column matrix or column vector (2.1.4)

A matrix with only one column is a column matrix or column vector. Examples:

In general, a column matrix with $m$ rows is

Square matrix (2.1.5)

A matrix with an equal number of rows and columns is a square matrix. If it has $n$ rows and $n$ columns, its order is $n\times n$. Examples:

Rectangular matrix (2.1.6)

If the number of rows is not equal to the number of columns ($m\ne n$), the matrix is rectangular. Examples:

Zero or null matrix (2.1.7)

If all the elements of a matrix are zero, it is a null or zero matrix. It is usually written $O_{m\times n}$. Examples:

Diagonal matrix (2.1.8)

A square matrix in which all elements except those on the main diagonal are zero, and the main diagonal has at least one non-zero element, is a diagonal matrix. The main diagonal runs from the top-left to the bottom-right corner ($a_{11},a_{22},\dots ,a_{nn}$).

In symbols, $A=[a_{ij}{]}_{n\times n}$ is diagonal if $a_{ij}=0$ for $i\ne j$ and $a_{ij}\ne 0$ for at least one $i=j$, where $i,j=1,2,3,\dots ,n$. Examples:

Scalar matrix (2.1.9)

A diagonal matrix in which all diagonal elements are the same non-zero number is a scalar matrix. That is, for $A=[a_{ij}{]}_{n\times n}$:

Identity or unit matrix (2.1.10)

A scalar matrix in which all the diagonal elements are 1 is the identity matrix or unit matrix. It is written $I_{n\times n}$ or simply $I$. For $I=[a_{ij}]$:

The $n\times n$ identity matrix is written $I_n$:

Upper triangular matrix (2.1.11)

A square matrix in which all elements below the main diagonal are zero is an upper triangular matrix. That is, $A=[a_{ij}{]}_{n\times n}$ is upper triangular if $a_{ij}=0$ whenever $i>j$. Examples:

Lower triangular matrix (2.1.13)

A square matrix in which all elements above the main diagonal are zero is a lower triangular matrix. That is, $A=[a_{ij}{]}_{n\times n}$ is lower triangular if $a_{ij}=0$ whenever $i<j$. Examples:

Triangular matrix (2.1.14)

A square matrix that is either upper triangular or lower triangular is called a triangular matrix.

2.1.15 to 2.1.17Transpose, symmetric and skew-symmetric matricesOften examined

Transpose of a matrix (2.1.15)

If A is a matrix of order m\times n, the matrix obtained by interchanging its rows and columns (rows become columns, or columns become rows) is the transpose of A, denoted A^{t}. The order of A^{t} is n\times m.

For example, if

The first row (2,1,6) of A became the first column of A^t, and the second row (0,2,3) became its second column.

Symmetric matrix (2.1.16)

A square matrix A is symmetric if A=A^{t}. For example,

Since A=A^{t}, the matrix A is symmetric. The matrix is a mirror image of itself across the main diagonal.

Skew-symmetric matrix (2.1.17)

A square matrix A is skew-symmetric if A=-A^{t}. For example, for

we get

So A is skew-symmetric.

2.2, 2.2.1 to 2.2.4Algebra of matricesOften examined

Algebra of matrices

Scalar multiplication (2.2.1)

If k is a non-zero scalar and A=[a_{ij}]_{m\times n} is a matrix of order m\times n, the product kA is the matrix obtained by multiplying every element of A by k.

For example, if A=\begin{bmatrix}3 & 2 & 1 \\ 4 & -3 & 5\end{bmatrix}, then

Addition of matrices (2.2.2)

In general we cannot add two matrices. Only matrices of the same order are conformable for addition. If A=[a_{ij}]_{m\times n} and B=[b_{ij}]_{m\times n}, then A+B is also a matrix of order m\times n, in which each element is the sum of the corresponding elements of A and B. If A+B=C with C=[c_{ij}]_{m\times n}, then

Subtraction of matrices (2.2.3)

Similarly, two matrices of the same order can be subtracted. If A=[a_{ij}]_{m\times n} and B=[b_{ij}]_{m\times n}, then A-B is a matrix of order m\times n in which each element is the difference of the corresponding elements. If A-B=D with D=[d_{ij}]_{m\times n}, then

Multiplication of matrices (2.2.4)

Two matrices A and B are conformable for multiplication (in the order AB) if the **number of columns of A equals the number of rows of B**. If A=[a_{ij}]_{m\times n} and B=[b_{ij}]_{n\times p}, then AB has order m\times p.

Let AB=C=[c_{ij}]_{m\times p}. The element c_{ij} is the sum of the products of the elements of the ith row of A with the corresponding elements of the jth column of B:

2.2.5 and 2.2.6Commutative property of matrices

Commutative property with respect to addition (2.2.5)

Any two matrices that are conformable for addition satisfy the commutative property with respect to addition. Take A=[a_{ij}]_{2\times3} and B=[b_{ij}]_{2\times3} (real entries). Then

Each entry is a sum of two real numbers, and real numbers can be added in either order (a+b=b+a). So

Hence matrix addition is commutative: A+B=B+A.

Commutative property with respect to multiplication (2.2.6)

In general, matrix multiplication is not commutative: usually AB\neq BA. Sometimes BA does not even exist when AB does, and even when both exist they may have different orders or different entries.

2.2.6 (Examples 6 to 9)Transpose rules and symmetric and skew-symmetric partsOften examined

The next four examples of the book use the transpose. They show how the transpose behaves with products and sums, and that every square matrix splits into a symmetric part and a skew-symmetric part.

2.3, 2.3.1 and 2.3.2DeterminantsOften examined

Every square matrix is associated with a number (real or complex) called its determinant. If A is a square matrix, its determinant is written \det(A) or |A|. For a square matrix A of order n, the determinant is written by replacing the square brackets by vertical bars:

Determinant of a matrix of order 2 × 2 (2.3.1)

For A=\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix},

Determinant of a matrix of order 3 × 3 (2.3.2)

For A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix} the associated determinant is |A|=\begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}.

To find its value, we write the determinant as a sum or difference of determinants of order 2. This process is called the expansion of the determinant. A determinant of order 3 has three rows R_1,R_2,R_3 and three columns C_1,C_2,C_3, so it can be expanded from any row or any column: six different ways, and the value is the same in every case.

Expanding along the first row R_1:

This can be generalised to determinants of higher order.

2.3.3 and 2.3.4Minors and cofactorsOften examined

Minor of an element (2.3.3)

Let A=[a_{ij}]_{n\times n} be a square matrix of order n. The minor of the element a_{ij} is the determinant of order (n-1)\times(n-1) obtained by **neglecting (deleting) the ith row and the jth column** of A. It is denoted M_{ij}.

For example, for A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}, the minor of a_{21} is obtained by neglecting the 2nd row and the 1st column:

In the same way we can find all the minors of the elements of A.

Cofactor of an element (2.3.4)

For a square matrix A of order n\times n, the cofactor of the element a_{ij} is denoted A_{ij} and defined by

For example, for the same matrix A the cofactor of a_{21} is

2.3.5 and 2.3.6Evaluation of a determinant by cofactors, singular matricesOften examined

Evaluation of a determinant using cofactors (2.3.5)

Take a square matrix A of order 3\times3, with |A|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}. Expand from the first row. The expansion of the previous topic gives

If we expand from the first column instead:

So |A| can be evaluated by adding the products of the elements of any row (or column) with their corresponding cofactors.

Singular and non-singular matrices (2.3.6)

A square matrix A is singular if |A|=0. If |A|\neq0, it is called a non-singular matrix.

For example, take A=\begin{bmatrix}1 & 3 \\ 4 & 2\end{bmatrix} and B=\begin{bmatrix}3 & 12 \\ \tfrac12 & 2\end{bmatrix}. Then

Thus A is a non-singular matrix and B is a singular matrix.

2.3.4 (adjoint), 2.3.7 and 2.3.8Adjoint and inverse of a matrixOften examined

Adjoint of a square matrix (2.3.4 in the book, adjoint)

Take any square matrix A=[a_{ij}] of order n. The adjoint of A, written \operatorname{adj}(A), is the matrix whose entries are the cofactors of A, arranged in the transposed order: the cofactor of a_{ij} goes in row j, column i.

For a matrix of order 3\times3:

Multiplicative inverse of a square matrix (2.3.7)

Two square matrices of the same order n are multiplicative inverses of each other if their product is I_n, the identity matrix of order n. Only non-singular matrices have a multiplicative inverse. The inverse of a non-singular matrix A is denoted A^{-1} and

Adjoint method to find the inverse (2.3.8)

If A is a non-singular square matrix, that is |A|\neq0, then

If A is singular, |A|=0, so \dfrac{1}{|A|}\operatorname{adj}(A) does not exist and A^{-1} does not exist.

2.4 (Properties 1 to 4)Properties of determinants: transpose, interchange, identical rows, scalarOften examined

Here we discuss some important properties of determinants which help us to find the value of a determinant. For convenience, the book takes a square matrix A of order 3\times3:

Property 1: determinant of the transpose

For any square matrix A, the determinant of the matrix and the determinant of its transpose are the same: |A|=|A^{t}|.

Property 2: interchanging two rows or columns

If any two rows (or columns) of a square matrix A are interchanged, and the resulting matrix is B, then |B|=-|A|.

Property 3: two identical rows or columns

If any two rows (or columns) of a square matrix are identical, the value of the determinant is zero.

Property 4: multiplying a row or column by a scalar

If each element of a row (or column) is multiplied by a non-zero scalar k, and the resulting matrix is B, then |B|=k|A|. In other words, a common factor of one row (or column) can be taken out of the determinant.

2.4 (Properties 5 to 8)Properties of determinants: sums, zero rows, row operations, triangular matricesOften examined

Property 5: a row or column of sums

If each element of one column (or row) is the sum of two numbers, the determinant equals the sum of two determinants. The first has the first part of each sum in that column (row) and the remaining columns (rows) unchanged. The second has the second part of each sum in that column (row) and the remaining columns (rows) unchanged.

Property 6: a row or column of zeros

If all the elements of a row (or column) of a square matrix A are zero, then |A|=0.

Property 7: adding a multiple of a row or column

If we multiply any row (column) of a square matrix by a scalar k and add the result to the corresponding elements of any other row (column), the value of the determinant is unchanged.

Property 8: triangular and diagonal matrices

If a square matrix A is upper triangular, lower triangular or diagonal, then |A| is the product of its diagonal elements.

2.4.1Evaluation of determinants without expansionOften examined

Evaluation of determinants without expansion (2.4.1)

The properties let us find the value of a determinant, or prove that it has a given value, without expanding it. The usual plan is: use Property 7 (row and column operations) to create a zero row or column, or two identical rows, or to make a common factor, then apply Property 3, 4 or 6.

2.5 (2.5.1 to 2.5.5)Row and column operations on matricesOften examined

Row and column operations on matrices (2.5.1)

Elementary row operations

The following elementary row operations can be performed on a matrix:

1. Interchange any two rows. Interchanging the ith and jth rows is denoted R_{ij}.
2. Multiply any row by a non-zero scalar k. Multiplying the ith row by k is denoted kR_i.
3. Add a multiple of any row to the corresponding elements of any other row. Adding k times the jth row to the ith row is denoted R_i+kR_j.

Elementary column operations

In the same way, the elementary column operations are:

1. Interchange any two columns: the ith with the jth column is denoted C_{ij}.
2. Multiply any column by a non-zero scalar k: denoted kC_i.
3. Add k times the jth column to the ith column: denoted C_i+kC_j.

Echelon form of a matrix (2.5.2)

A matrix is in row echelon form if:

1. the first non-zero element in each row, called the leading entry, is 1;
2. each leading entry is in a column to the right of the leading entry in the previous row;
3. any row which has all elements zero is below the rows having at least one non-zero element.

Reduced echelon form of a matrix (2.5.3)

A matrix is in row reduced echelon form if:

1. it satisfies the conditions of row echelon form, and
2. the leading entry in each row is the only non-zero entry in its column.

Rank of a matrix (2.5.4)

To find the rank of a matrix, reduce it to its echelon form (or reduced echelon form). The number of non-zero rows is called the rank (or row rank) of the matrix.

Using row operations to find the inverse (2.5.5)

Row operations can be used on a non-singular matrix A to find its inverse. Take the identity matrix I of the same order as A and write A and I side by side, as [A:I]. Perform the same row operations on both, so that A is reduced to I. Then I is reduced to a new matrix, which is A^{-1}:

2.6 and 2.6.1Linear equations and systems of linear equations

Linear equation

An equation of the form a_1x_1+a_2x_2=k_1, where a_1,a_2 and k_1 are constants and at least one of a_1 and a_2 is non-zero, is a linear equation in two variables x_1 and x_2.

Similarly, an equation of the form a_1x_1+a_2x_2+a_3x_3=k_2, where a_1,a_2,a_3 and k_2 are constants and at least one of a_1,a_2,a_3 is non-zero, is a linear equation in three variables x_1,x_2,x_3. In the same manner this extends to n variables.

System of linear equations

When we deal with more than one linear equation at the same time, we have a system of linear equations. A system is divided into two categories:

1. homogeneous system of linear equations;
2. non-homogeneous system of linear equations.

Homogeneous and non-homogeneous linear equations (2.6.1)

Homogeneous system

Consider the system of three linear equations in three unknowns

If k_1=k_2=k_3=0, the system is called a homogeneous system of linear equations:

Non-homogeneous system

For the same system, if at least one of k_1,k_2,k_3 is non-zero, the system is called a non-homogeneous system of linear equations.

2.6.2Solution of a homogeneous system of linear equationsOften examined

Solution of a system of linear equations (2.6.2)

The values of the variables which, when substituted in any equation of the system, satisfy that equation are the solution of the system. A system may have no solution, a unique solution, or infinitely many solutions.

Solution of a homogeneous system

Consider the homogeneous system

In matrix form it is written

Let A be the matrix of coefficients, X the column of unknowns and O the zero column. Then the system is

Each equation is satisfied by x=0,\;y=0,\;z=0. So (0,0,0) is always a solution of a homogeneous system. Because this solution always exists, it is called the trivial solution. Every solution other than the trivial one is a non-trivial solution.

2.6.3 to 2.6.8Consistent and inconsistent systems, consistency criteriaOften examined

Consistent system of equations (2.6.3)

A system of linear equations which has at least one solution is a consistent system of equations.

Inconsistent system of equations (2.6.4)

A system of linear equations which has no solution at all is an inconsistent system of equations.

Solution of a non-homogeneous system (2.6.5)

Consider the non-homogeneous system

where at least one of k_1,k_2,k_3 is non-zero. In matrix form

Let A=\begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix}, X=\begin{bmatrix}x\\y\\z\end{bmatrix} and B=\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}. Then the system is

Augmented matrix (2.6.7)

For a system of linear equations, the matrix made of the coefficients of the unknowns together with the constants on the right side is the augmented matrix. It is usually denoted A_b. For the system above,

Consistency criteria (2.6.6)

A system of linear equations is consistent if \operatorname{Rank}A=\operatorname{Rank}A_b, and inconsistent if \operatorname{Rank}A\neq\operatorname{Rank}A_b. When the system is consistent:

• if \operatorname{Rank}A=\operatorname{Rank}A_b= number of unknowns, the system has a unique solution;
• if \operatorname{Rank}A=\operatorname{Rank}A_b< number of unknowns, the system has infinitely many solutions.

Methods to solve a non-homogeneous system (2.6.8)

To solve a non-homogeneous system of linear equations of order 3\times3, the book uses four methods:

• Matrix inversion method
• Gauss elimination method (echelon form)
• Gauss-Jordan method (reduced echelon form)
• Cramer's rule

2.6.8 (matrix inversion method)Solving a system by the matrix inversion methodOften examined

Matrix inversion method

Consider the non-homogeneous system a_1x+b_1y+c_1z=k_1,\;a_2x+b_2y+c_2z=k_2,\;a_3x+b_3y+c_3z=k_3. In matrix form it is AX=B.

2.6.8 (Gauss elimination, Gauss-Jordan)Gauss elimination and Gauss-Jordan methodsOften examined

Gauss elimination method (echelon form)

In this method we reduce the augmented matrix A_b of the system to its echelon form by row operations. Then we solve from the last row upwards (back substitution): the last row gives one unknown, the row above gives the next, and so on.

Gauss-Jordan method (reduced echelon form)

In this method we reduce the augmented matrix to its reduced echelon form. The reduced form gives the values of the unknowns directly (or the relations between them), without back substitution.

2.6.8 (Cramer's rule)Cramer's ruleOften examined

Cramer's rule

Consider the non-homogeneous system a_1x+b_1y+c_1z=k_1,\;a_2x+b_2y+c_2z=k_2,\;a_3x+b_3y+c_3z=k_3, written AX=B. If A is non-singular, |A|\neq0 and A^{-1} exists.

2.7Application of matrices

Cryptography: encoding and decoding a message

Matrices are used in many disciplines. In cryptography a message is scrambled (encoded) so that only the person with the key can read it. The book explains encryption and decryption by an example.

Suppose the sender and the receiver consider messages in the letters A to Z only. Both assign the numbers 1 to 26 to the letters A to Z respectively, and the number 0 to a blank space. The sender uses as the key post-multiplication by a non-singular matrix of order 3 of his own choice. The receiver decodes by post-multiplication by the inverse of the matrix chosen by the sender.

Let the encoding matrix be

Let the message be "WELCOME". Since the key is a matrix of order 3, the message is cut into pieces of length 3: (WEL), (COM), (E), and converted into row matrices of numbers using W=23, E=5, L=12, C=3, O=15, M=13:

Two zeros (blanks) are added to the last row matrix so that every piece has length 3 and its first entry is 5 (the letter E).

Encoding

Post-multiply each row matrix by A (row times matrix gives a row).

Decoding

The receiver post-multiplies by the inverse of A. First find A^{-1} by the adjoint method. Here |A|=1 and the cofactors give

Transformation of a graph by a matrix

Matrices can also describe a shift of a figure in the plane. The following example, from the book, moves a line one unit up.

Formula sheet

Key terms

Matrix

An array of numbers arranged in rows and columns and enclosed in square brackets.

Order of a matrix

The number of rows by the number of columns, m\times n, rows first.

Element (entry)

Each number in a matrix. The element in row i and column j is a_{ij}.

Equal matrices

Matrices of the same order with the same corresponding elements.

Row matrix and column matrix

A matrix with only one row, or only one column.

Square matrix

A matrix with equal numbers of rows and columns.

Zero (null) matrix

A matrix all of whose elements are zero.

Diagonal matrix

A square matrix in which every element off the main diagonal is zero and at least one diagonal element is non-zero.

Scalar matrix

A diagonal matrix whose diagonal elements are all the same non-zero number k.

Identity (unit) matrix

A scalar matrix whose diagonal elements are all 1, written I.

Triangular matrix

A square matrix that is upper triangular (zeros below the diagonal) or lower triangular (zeros above the diagonal).

Transpose

The matrix A^{t} obtained by interchanging the rows and columns of A.

Symmetric matrix

A square matrix with A=A^{t}.

Skew-symmetric matrix

A square matrix with A=-A^{t}.

Conformable matrices

Matrices that can be added (same order) or multiplied (columns of the first equal rows of the second).

Determinant

The number |A| or \det(A) associated with a square matrix A.

Minor

The determinant M_{ij} left after deleting the ith row and jth column.

Cofactor

A_{ij}=(-1)^{i+j}M_{ij}, the signed minor of a_{ij}.

Singular and non-singular matrix

A square matrix is singular if |A|=0 and non-singular if |A|\neq0.

Adjoint

\operatorname{adj}(A), the transpose of the matrix of cofactors of A.

Multiplicative inverse

The matrix A^{-1} with AA^{-1}=A^{-1}A=I; it exists only for a non-singular matrix.

Elementary row operations

Interchanging two rows (R_{ij}), multiplying a row by a non-zero scalar (kR_i), adding a multiple of a row to another row (R_i+kR_j).

Echelon form

A matrix whose leading entries are 1, move to the right row by row, and whose zero rows are at the bottom.

Reduced echelon form

An echelon form in which each leading entry 1 is the only non-zero entry in its column.

Rank

The number of non-zero rows in the echelon form of a matrix.

Homogeneous system

A system of linear equations in which all the constants on the right side are zero.

Non-homogeneous system

A system of linear equations in which at least one constant on the right side is non-zero.

Trivial and non-trivial solution

The solution x=y=z=0 of a homogeneous system is trivial; any other solution is non-trivial.

Consistent and inconsistent system

A system with at least one solution is consistent; a system with no solution is inconsistent.

Augmented matrix

The matrix A_b made of the coefficients of the unknowns together with the constants on the right side.

Cramer's rule

The solution x=D_x/|A|, y=D_y/|A|, z=D_z/|A| of AX=B with |A|\neq0.