Matrices for CUET Maths

Q: When are two matrices equal?

Two matrices are equal when they have the same order and every pair of corresponding elements is equal. So one matrix equation is really several ordinary equations, one for each place.

Q: When can two matrices be multiplied?

AB exists only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p: the inner numbers must match, and the outer numbers give the order.

Q: Is matrix multiplication commutative?

No. In general AB and BA are different matrices, and they can differ in three ways. One of them may not be defined at all: a 2 × 3 times a 3 × 4 works, the other order does not. Both may be defined but of different orders: a 2 × 3 and a 3 × 2 give 2 × 2 one way and 3 × 3 the other. Or, as above, both have the same order and are still different.

Q: What is the transpose of a matrix?

The transpose A', also written A^T, turns the rows of A into columns, in order: row 1 becomes column 1. If A is m × n, then A' is n × m, and the element in row i, column j of A' is aⱼᵢ.

Q: What makes a matrix symmetric?

A square matrix is symmetric when A' = A, that is aᵢⱼ = aⱼᵢ for every i and j: it is its own mirror image across the leading diagonal. It is skew-symmetric when A' = -A, that is aⱼᵢ = -aᵢⱼ.

Q: When is a matrix invertible?

A square matrix A is invertible when there is a matrix B of the same order with AB = BA = I. Then B is the inverse of A, written A^-1, and A is the inverse of B as well.

About 10 min to learn · 106 questions

Matrices are part of Section A of CUET Maths, the section every candidate answers, whether you take Mathematics or Applied Mathematics. The questions are short and rule-based, so the marks come quickly once the rules are clear. Learn each rule here, from orders and types to the transpose and the inverse, with worked examples, time-saving shortcuts and the traps CUET sets, then practise on free questions in CUET's own formats, each fully solved.

Syllabus area: Section A1

On this page · 14 sections

01What a matrix is
02Types of matrices
03Equality of matrices
04Adding matrices and multiplying by a number
05Multiplying two matrices
06Properties of multiplication
07The transpose
08Symmetric and skew-symmetric matrices
09Invertible matrices
10How CUET asks this chapter
11Exam shortcuts
12Common mistakes
13Quick revision
14Check your understanding

What a matrix is

A stationery shop has two branches. On Monday, branch 1 sells 12 pens, 5 notebooks and 8 erasers, and branch 2 sells 9 pens, 7 notebooks and 4 erasers. Written as a table with one row per branch and one column per item, the numbers are

S = \begin{bmatrix} 12 & 5 & 8 \\ 9 & 7 & 4 \end{bmatrix}

That rectangle of numbers is a matrix, and each number in it is an element. Rows run across and columns run down. We will come back to this shop when we add and multiply matrices.

What is the order of a matrix?

The order is the number of rows by the number of columns, written $m \times n$ , rows first. $S$ has 2 rows and 3 columns, so its order is $2 \times 3$ , and it has $2 \times 3 = 6$ elements.

The element in row $i$ and column $j$ of a matrix $A$ is written $a_{ij}$ , again row first. In $S$ , $s_{12} = 5$ is branch 1's notebooks and $s_{21} = 9$ is branch 2's pens.

Example 1 · How many orders?

A matrix has 18 elements. What orders can it have?

An order $m \times n$ needs $mn = 18$ , so $m$ can be any divisor of 18: 1, 2, 3, 6, 9 or 18.
Each divisor gives one order, and $2 \times 9$ is a different order from $9 \times 2$ .

Answer

Six orders: $1 \times 18$ , $2 \times 9$ , $3 \times 6$ , $6 \times 3$ , $9 \times 2$ and $18 \times 1$ .

A rule can also give the elements. " $A = [a_{ij}]$ is $2 \times 3$ with $a_{ij} = 3i - j$ " means: put each row number $i$ and column number $j$ into the rule.

Example 2 · A matrix from a rule

Write the $2 \times 3$ matrix $A = [a_{ij}]$ with $a_{ij} = 3i - j$ .

Row 1 ( $i = 1$ ): $3 - 1 = 2$ , $3 - 2 = 1$ and $3 - 3 = 0$ .
Row 2 ( $i = 2$ ): $6 - 1 = 5$ , $6 - 2 = 4$ and $6 - 3 = 3$ .

Answer

A = \begin{bmatrix} 2 & 1 & 0 \\ 5 & 4 & 3 \end{bmatrix}

Key idea

To count matrices, count places. An $m \times n$ matrix has $mn$ places, and if each can take any of $r$ values there are $r^{mn}$ matrices. A $2 \times 2$ matrix of 0s and 1s can be made in $2^4 = 16$ ways.

Types of matrices

What are the types of matrices?

Seven types come up again and again: row, column, square, diagonal, scalar, identity and zero matrices.

Type	What it is	Example
Row matrix	one row, order $1 \times n$	$\begin{bmatrix} 4 & -2 & 7 \end{bmatrix}$
Column matrix	one column, order $m \times 1$	$\begin{bmatrix} 4 \\ -2 \end{bmatrix}$
Square matrix	as many rows as columns	$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
Diagonal matrix	square, with 0 everywhere off the leading diagonal	$\begin{bmatrix} 2 & 0 \\ 0 & -1 \end{bmatrix}$
Scalar matrix	diagonal, with all the diagonal elements equal	$\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$
Identity matrix $I$	diagonal, with every diagonal element 1	$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
Zero matrix $O$	every element 0, of any order	$\begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$

The leading diagonal of a square matrix runs from the top left to the bottom right: $a_{11}, a_{22}, a_{33}$ and so on.

Watch out

The types nest: every identity matrix is a scalar matrix, every scalar matrix is diagonal, and every diagonal matrix is square. None of these works backwards. A zero matrix need not be square.

Equality of matrices

When are two matrices equal?

Two matrices are equal when they have the same order and every pair of corresponding elements is equal. So one matrix equation is really several ordinary equations, one for each place.

Example 3 · Unknowns from equal matrices

Find $x$ , $y$ and $z$ if $\begin{bmatrix} x + y & 2 \\ x - y & z - 1 \end{bmatrix} = \begin{bmatrix} 9 & 2 \\ 5 & 3 \end{bmatrix}$ .

Corresponding elements give $x + y = 9$ , $x - y = 5$ and $z - 1 = 3$ .
Adding the first two, $2x = 14$ , so $x = 7$ and $y = 2$ . The third gives $z = 4$ .

Answer

$x = 7$ , $y = 2$ and $z = 4$

If the equations contradict each other, say one place gives $x = 3$ and another $x = 0$ , then no values make the matrices equal, and "no such values" is the answer.

Adding matrices and multiplying by a number

Matrices of the same order are added element by element; matrices of different orders cannot be added at all. Multiplying a matrix by a number $k$ , called a scalar, multiplies every element by $k$ . Then $-A = (-1)A$ and $A - B = A + (-B)$ .

Back at the shop, suppose Tuesday's sales are $T = \begin{bmatrix} 10 & 6 & 5 \\ 8 & 4 & 6 \end{bmatrix}$ . The two days together are

S + T = \begin{bmatrix} 22 & 11 & 13 \\ 17 & 11 & 10 \end{bmatrix}

so branch 2 sold 17 pens over the two days. If a festival doubled Monday's sales, they would be $2S$ .

Properties of addition and scalar multiplication

A + B = B + A, \qquad (A + B) + C = A + (B + C), \qquad A + O = A, \qquad A + (-A) = O

k(A + B) = kA + kB, \qquad (k + l)A = kA + lA

Example 4 · Two matrix equations at once

Find $X$ and $Y$ if $X + Y = \begin{bmatrix} 6 & 2 \\ 4 & 8 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 2 & 0 \\ -2 & 4 \end{bmatrix}$ .

Add the two equations: $2X = \begin{bmatrix} 8 & 2 \\ 2 & 12 \end{bmatrix}$ , so $X = \begin{bmatrix} 4 & 1 \\ 1 & 6 \end{bmatrix}$ .
Subtract them: $2Y = \begin{bmatrix} 4 & 2 \\ 6 & 4 \end{bmatrix}$ , so $Y = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$ .

Answer

$X = \begin{bmatrix} 4 & 1 \\ 1 & 6 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$

Multiplying two matrices

When can two matrices be multiplied?

$AB$ exists only when the number of columns of $A$ equals the number of rows of $B$ . If $A$ is $m \times n$ and $B$ is $n \times p$ , then $AB$ is $m \times p$ : the inner numbers must match, and the outer numbers give the order.

Each element of $AB$ is a row of $A$ times a column of $B$ : multiply the matching entries and add them up. The shop shows why this is the natural rule. If a pen costs ₹10, a notebook ₹40 and an eraser ₹5, put the prices in a column $P$ :

SP = \begin{bmatrix} 12 & 5 & 8 \\ 9 & 7 & 4 \end{bmatrix}\begin{bmatrix} 10 \\ 40 \\ 5 \end{bmatrix} = \begin{bmatrix} 12(10) + 5(40) + 8(5) \\ 9(10) + 7(40) + 4(5) \end{bmatrix} = \begin{bmatrix} 360 \\ 390 \end{bmatrix}

Row 1 times the price column is branch 1's takings, ₹360, and row 2 is branch 2's, ₹390. A $2 \times 3$ matrix times a $3 \times 1$ matrix gave a $2 \times 1$ matrix.

Example 5 · AB and BA

Find $AB$ and $BA$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 1 \\ 2 & 5 \end{bmatrix}$ .

$AB$ : row 1 gives $1(4) + 2(2) = 8$ and $1(1) + 2(5) = 11$ ; row 2 gives $3(4) + 0(2) = 12$ and $3(1) + 0(5) = 3$ .
$BA$ : row 1 gives $4(1) + 1(3) = 7$ and $4(2) + 1(0) = 8$ ; row 2 gives $2(1) + 5(3) = 17$ and $2(2) + 5(0) = 4$ .

Answer

$AB = \begin{bmatrix} 8 & 11 \\ 12 & 3 \end{bmatrix}$ but $BA = \begin{bmatrix} 7 & 8 \\ 17 & 4 \end{bmatrix}$ , so $AB \ne BA$ .

Is matrix multiplication commutative?

No. In general $AB$ and $BA$ are different matrices, and they can differ in three ways. One of them may not be defined at all: a $2 \times 3$ times a $3 \times 4$ works, the other order does not. Both may be defined but of different orders: a $2 \times 3$ and a $3 \times 2$ give $2 \times 2$ one way and $3 \times 3$ the other. Or, as above, both have the same order and are still different.

Watch out

A product can be the zero matrix when neither factor is: $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ . So $AB = O$ does not mean $A = O$ or $B = O$ , and $AB = AC$ does not mean $B = C$ .

Properties of multiplication

Matrix multiplication keeps three rules of ordinary multiplication and loses one.

What multiplication keeps

(AB)C = A(BC), \qquad A(B + C) = AB + AC, \qquad (A + B)C = AC + BC, \qquad AI = IA = A

The rule it loses is $AB = BA$ . Because of that, $(A + B)^2 = A^2 + AB + BA + B^2$ , which is $A^2 + 2AB + B^2$ only when $A$ and $B$ commute.

Powers make sense for a square matrix: $A^2 = AA$ and $A^3 = A^2A$ . In a matrix polynomial such as $A^2 - 5A + 6I$ , the constant term is that number times $I$ , so that every term is a matrix of the same order.

Example 6 · A matrix polynomial

For $A = \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix}$ , show that $A^2 - 5A + 6I = O$ .

$A^2 = \begin{bmatrix} 16 - 2 & 4 + 1 \\ -8 - 2 & -2 + 1 \end{bmatrix} = \begin{bmatrix} 14 & 5 \\ -10 & -1 \end{bmatrix}$ .
$5A = \begin{bmatrix} 20 & 5 \\ -10 & 5 \end{bmatrix}$ , so $A^2 - 5A = \begin{bmatrix} -6 & 0 \\ 0 & -6 \end{bmatrix}$ .
Adding $6I$ clears the diagonal.

Answer

A^2 - 5A + 6I = O

Key idea

Every $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ satisfies $A^2 - (a + d)A + (ad - bc)I = O$ . In the example $a + d = 5$ and $ad - bc = 4 + 2 = 6$ . So when a question asks for $k$ in $A^2 - kA + cI = O$ , $k$ is the sum of the diagonal elements. The next chapter calls $ad - bc$ the determinant.

Two more patterns come up often. If $A^2 = A$ , every power of $A$ is $A$ , so $(I + A)^2 = I + 2A + A^2 = I + 3A$ . And if $A = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$ , then $A^n = \begin{bmatrix} 1 & nk \\ 0 & 1 \end{bmatrix}$ .

The transpose

What is the transpose of a matrix?

The transpose $A'$ , also written $A^T$ , turns the rows of $A$ into columns, in order: row 1 becomes column 1. If $A$ is $m \times n$ , then $A'$ is $n \times m$ , and the element in row $i$ , column $j$ of $A'$ is $a_{ji}$ .

A = \begin{bmatrix} 2 & 7 & -1 \\ 0 & 3 & 6 \end{bmatrix} \quad \Rightarrow \quad A' = \begin{bmatrix} 2 & 0 \\ 7 & 3 \\ -1 & 6 \end{bmatrix}

Properties of the transpose

(A')' = A, \qquad (kA)' = kA', \qquad (A + B)' = A' + B', \qquad (AB)' = B'A'

Watch out

The order reverses in $(AB)' = B'A'$ . Writing $A'B'$ is the most common slip, and it can fail outright: if $A$ is $2 \times 3$ and $B$ is $3 \times 4$ , then $(AB)'$ is $4 \times 2$ , but $A'B'$ is not even defined.

Symmetric and skew-symmetric matrices

What makes a matrix symmetric?

A square matrix is symmetric when $A' = A$ , that is $a_{ij} = a_{ji}$ for every $i$ and $j$ : it is its own mirror image across the leading diagonal. It is skew-symmetric when $A' = -A$ , that is $a_{ji} = -a_{ij}$ .

For example, $\begin{bmatrix} 1 & 4 & -2 \\ 4 & 0 & 5 \\ -2 & 5 & 3 \end{bmatrix}$ is symmetric and $\begin{bmatrix} 0 & 4 & -2 \\ -4 & 0 & 5 \\ 2 & -5 & 0 \end{bmatrix}$ is skew-symmetric.

Putting $j = i$ in $a_{ji} = -a_{ij}$ gives $a_{ii} = -a_{ii}$ , so $2a_{ii} = 0$ : every diagonal element of a skew-symmetric matrix is 0.

For any square matrix $A$ , the matrix $A + A'$ is symmetric, because $(A + A')' = A' + A$ , and $A - A'$ is skew-symmetric, because $(A - A')' = A' - A = -(A - A')$ . Their halves add up to $A$ , so every square matrix is the sum of a symmetric matrix and a skew-symmetric one.

Splitting any square matrix

A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')

The first part is symmetric and the second skew-symmetric.

Example 7 · The two parts

Write $A = \begin{bmatrix} 2 & 7 \\ 1 & 4 \end{bmatrix}$ as $P + Q$ , with $P$ symmetric and $Q$ skew-symmetric.

$A + A' = \begin{bmatrix} 4 & 8 \\ 8 & 8 \end{bmatrix}$ , so $P = \begin{bmatrix} 2 & 4 \\ 4 & 4 \end{bmatrix}$ .
$A - A' = \begin{bmatrix} 0 & 6 \\ -6 & 0 \end{bmatrix}$ , so $Q = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}$ .

Answer

$P + Q = \begin{bmatrix} 2 & 7 \\ 1 & 4 \end{bmatrix} = A$ , with $P' = P$ and $Q' = -Q$ .

The same one-line test, take the transpose and compare, settles the facts that are asked most:

a sum, difference or scalar multiple of symmetric matrices is symmetric, and the same holds for skew-symmetric matrices;
the only matrix that is both symmetric and skew-symmetric is $O$ , since $A' = A$ and $A' = -A$ give $A = -A$ ;
if $A$ and $B$ are symmetric, $AB$ is symmetric only when $AB = BA$ , while $AB - BA$ is always skew-symmetric;
$AA'$ and $A'A$ are always symmetric.

Invertible matrices

When is a matrix invertible?

A square matrix $A$ is invertible when there is a matrix $B$ of the same order with $AB = BA = I$ . Then $B$ is the inverse of $A$ , written $A^{-1}$ , and $A$ is the inverse of $B$ as well.

Example 8 · Checking an inverse

Show that $B = \begin{bmatrix} 1 & -2 \\ -2 & 5 \end{bmatrix}$ is the inverse of $A = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}$ .

$AB = \begin{bmatrix} 5 - 4 & -10 + 10 \\ 2 - 2 & -4 + 5 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ .
$BA = \begin{bmatrix} 5 - 4 & 2 - 2 \\ -10 + 10 & -4 + 5 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ as well.

Answer

A^{-1} = \begin{bmatrix} 1 & -2 \\ -2 & 5 \end{bmatrix}

Three facts follow from the definition:

only a square matrix can have an inverse, because $AB$ and $BA$ must both be defined and of the same order;
an inverse is unique: if $B$ and $C$ are both inverses of $A$ , then $B = B(AC) = (BA)C = C$ ;
the inverse of a product reverses the order, $(AB)^{-1} = B^{-1}A^{-1}$ , just as the transpose does.

Not every square matrix is invertible; $O$ has no inverse, for one. A matrix equation can hand you the inverse directly: if $A^2 - 4A - I = O$ , then $A(A - 4I) = I$ and $(A - 4I)A = I$ , so $A^{-1} = A - 4I$ . Which matrices are invertible, and how to find $A^{-1}$ in general, is the next chapter's work.

How CUET asks this chapter

Orders and elements: the order of a product or a transpose, one element of a product, how many orders a matrix can have, or how many matrices can be made from given entries.
Unknowns: from two equal matrices, or from a symmetric or skew-symmetric condition.
Statement sets: which of several statements about products, transposes, symmetric or skew-symmetric matrices, or inverses are true.
Matrix polynomials: find $k$ in $A^2 - kA + cI = O$ , or work out a polynomial such as $A^2 - 5A + 6I$ .

Exam shortcuts

The number of orders for $N$ elements is the number of divisors of $N$ .
Matrices of order $m \times n$ with each element from $r$ values: $r^{mn}$ .
$AB$ is defined when the inner numbers match, and its order is the outer numbers.
For a $2 \times 2$ matrix, $A^2 - (a + d)A + (ad - bc)I = O$ .
If $A^2 = A$ , every power of $A$ is $A$ .
A skew-symmetric $3 \times 3$ matrix is fixed by its 3 elements above the diagonal.

Common mistakes

Reading $a_{ij}$ column first.
Multiplying matching entries and calling it matrix multiplication.
Assuming $AB = BA$ , or cancelling $A$ in $AB = AC$ .
Adding a plain number to a matrix: in $A^2 - 5A + 6I$ , the 6 is $6I$ , on the diagonal only.
Writing $(AB)' = A'B'$ or $(AB)^{-1} = A^{-1}B^{-1}$ .
Forgetting the $\frac{1}{2}$ in the symmetric and skew-symmetric parts.

Quick revision

Order and types

Order: rows × columns; $a_{ij}$ is row $i$ , column $j$
Identity, then scalar, then diagonal: each a special case of the next
Equal: same order and equal corresponding elements

Algebra

Add only matrices of the same order; $kA$ scales every element
$AB$ needs columns of $A$ = rows of $B$ ; $AB \ne BA$ in general
$AB = O$ can happen with $A \ne O$ and $B \ne O$

Transpose and symmetry

$(AB)' = B'A'$
Symmetric: $A' = A$ ; skew-symmetric: $A' = -A$ , zero diagonal
$A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')$

Inverse

$AB = BA = I$ , for square matrices only
Unique, and $(AB)^{-1} = B^{-1}A^{-1}$

The next chapter attaches a number to every square matrix and uses it to find inverses: determinants, the adjoint and the inverse.

Next · 13Quick revision

Check your understanding

106 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

A matrix, its elements and orderQ1 of 106Q1

A matrix, its elements and order

15 questions

List I	List II
A. $AB$	I. $2 \times 3$
B. $BA$	II. $2 \times 2$
C. $CA$	III. $3 \times 2$
D. $BC$	IV. $3 \times 3$

Check your understanding

A matrix, its elements and order

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Types of matrices

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Equality of matrices

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Addition and multiplication by a scalar

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Multiplication of matrices

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Properties of multiplication, and powers

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