(a) On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 5

Question

Q. (a) On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

(OR)

(b) Given that

P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}

Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}

and

R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}

, find a matrix

S

such that

PQ - RS

is a null matrix.

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Part (a)Concept understanding — Inverse Matrix Method

The Inverse Matrix Method

Many problems reduce to a system of linear equations, for example

\begin{aligned}2x+3y&=8\\x-\ \ y&=-1.\end{aligned}

The inverse matrix method solves such a system by writing it as a single matrix equation and then undoing the coefficient matrix with its inverse — the matrix analogue of dividing.

Writing the system as $AX=B$

Collect the coefficients, the unknowns and the constants:

A=\begin{pmatrix}2&3\\1&-1\end{pmatrix},\quad X=\begin{pmatrix}x\\y\end{pmatrix},\quad B=\begin{pmatrix}8\\-1\end{pmatrix},

so the whole system becomes $AX=B$ .

The idea

For numbers, $ax=b$ gives $x=a^{-1}b$ provided $a\neq0$ . The same works for matrices: if $A$ is invertible, multiply $AX=B$ on the left by $A^{-1}$ :

A^{-1}(AX)=A^{-1}B\;\Rightarrow\;IX=A^{-1}B\;\Rightarrow\;X=A^{-1}B.

X=A^{-1}B

Multiplying on the left matters — matrix products do not commute, so $BA^{-1}$ would be wrong.

When it works

The inverse $A^{-1}$ exists only when $\det A\neq0$ , so:

$\det A\neq0$ : the system is consistent with the unique solution $X=A^{-1}B$ .
$\det A=0$ : no inverse; the system is either inconsistent (no solution) or has infinitely many — handle it by another method.

Worked steps

For the system above, $\det A=(2)(-1)-(3)(1)=-5\neq0$ , and

A^{-1}=\frac{1}{-5}\begin{pmatrix}-1&-3\\-1&2\end{pmatrix}.

Then …

Part (b)Concept understanding — Matrix Equation Solving

Solving a System of Equations by the Matrix Method

A system of linear equations can be written as a single matrix equation and solved in one clean step using the inverse of a matrix. This is the Class-12 "matrix method" for simultaneous equations.

Writing the system as $AX=B$

Take the system

a_1x+b_1y+c_1z=d_1,\quad a_2x+b_2y+c_2z=d_2,\quad a_3x+b_3y+c_3z=d_3.

Collect the coefficients, the unknowns, and the constants into matrices:

A=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{pmatrix},\quad X=\begin{pmatrix}x\\y\\z\end{pmatrix},\quad B=\begin{pmatrix}d_1\\d_2\\d_3\end{pmatrix}.

Then the whole system is just

AX=B.

Solving when $A$ is invertible

If $\det(A)\neq0$ , then $A^{-1}$ exists, and multiplying both sides on the left by $A^{-1}$ gives

X=A^{-1}B,\qquad\text{where }\;A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A).

So you compute $\det(A)$ , then $\operatorname{adj}(A)$ , form $A^{-1}$ , and multiply by $B$ . The single column $X=A^{-1}B$ hands you $x$ , $y$ , $z$ at once, and because $A^{-1}$ is unique, the solution is unique.

Watch out

Multiply in the correct order: $X=A^{-1}B$ , not $BA^{-1}$ . Matrix multiplication is not commutative, and $BA^{-1}$ is not even defined here.

When $\det(A)=0$

If $\det(A)=0$ , $A^{-1}$ does not exist and the inverse method fails. The system is then either inconsistent (no solution) or has infinitely many solutions. Decide which by computing $(\operatorname{adj}A)\,B$ :

$(\operatorname{adj}A)\,B\neq O$ → no solution (inconsistent).
$(\operatorname{adj}A)\,B=O$ → infinitely many solutions (consistent, dependent). …

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The vouchers satisfy $x+y=60,\ x+3y=100$ ; solving $X=A^{-1}B$ gives 40 ₹1000 vouchers and 20 ₹500 vouchers.
$RS=PQ\Rightarrow S=R^{-1}(PQ)=\begin{bmatrix}-191&-110\\77&44\end{bmatrix}$ .

Part (a)

Let $x$ = number of ₹1000 vouchers and $y$ = number of ₹500 vouchers.

Total vouchers: $x+y=60$ .
" $x$ added to $3$ times $y$ gives $100$ ": $x+3y=100$ .

In matrix form $AX=B$ :

A=\begin{bmatrix}1&1\\1&3\end{bmatrix},\quad X=\begin{bmatrix}x\\y\end{bmatrix},\quad B=\begin{bmatrix}60\\100\end{bmatrix}.

$\det A=3-1=2\neq0$ , so $A^{-1}=\dfrac12\begin{bmatrix}3&-1\\-1&1\end{bmatrix}$ . Then

$X=A^{-1}B=\frac12\begin{bmatrix}3&-1\\-1&1\end{bmatrix}\begin{bmatrix}60\\100\end{bmatrix}=\frac12\begin{bmatrix}180-100\\-60+100\end{bmatrix}=\begin{bmatrix}40\\20\end{bmatrix}.$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 41 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$\begin{bmatrix} x+y \\ y+z \\ z+x \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$
(a) 9
(b) 0
(c) 4
(d) 5
›Reveal solutionSolution

Two matrices are equal only if all corresponding entries are equal; adding all three entry-equations gives $x+y+z$ directly.

From $\begin{bmatrix} x+y \\ y+z \\ z+x \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$ , equating corresponding entries:

…

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CBSE 2026Set ANNUAL1 markMCQ
$A = \begin{bmatrix} x & 0 \\ 2 & 3 \end{bmatrix}$
(a) $x = 4$
(b) $x = \pm3$
(c) $x = -3$
(d) $x = -4$
›Reveal solutionSolution

Computing $A^2$ and matching it to $9I$ forces both $x^2=9$ and $2x+6=0$ ; only $x=-3$ satisfies both.

$A=\begin{bmatrix}x & 0\\ 2 & 3\end{bmatrix}$ , so

$A^2=\begin{bmatrix}x & 0\\ 2 & 3\end{bmatrix}\begin{bmatrix}x & 0\\ 2 & 3\end{bmatrix}=\begin{bmatrix}x^2 & 0\\ 2x+6 & 9\end{bmatrix}$

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. If [[x-2y, 0], [5, x]] = [[-3, 0], [5, 3]], then y is equal to:$
(a) 1
(b) 3
(c) 2
(d) 4
›Reveal solutionSolution

Two matrices are equal only if all corresponding entries are equal; comparing the $(2,2)$ entries gives $x=3$ , then the $(1,1)$ entries give $y$ .

Given:

$\begin{bmatrix}x-2y & 0\\ 5 & x\end{bmatrix} = \begin{bmatrix}-3 & 0\\ 5 & 3\end{bmatrix}$

Comparing the $(2,2)$ entries: $x = 3$ .

…

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CBSE 2026Set ANNUAL1 markMCQ
$A$
(a) $\det(A)$
(b) $\frac{1}{\det(A)}$
(c) $1$
(d) $0$
›Reveal solutionSolution

$\det(A^{-1})=\dfrac{1}{\det A}$ .

From $AA^{-1}=I$ , $\det(A)\det(A^{-1})=\det(I)=1$ , so $\det(A^{-1})=\dfrac{1}{\det(A)}$ . (This holds for a …

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CBSE 2026Set ANNUAL1 mark
$AX = B$
›Reveal solutionSolution

The unique solution of $AX=B$ (when $|A| e0$ ) is $X=A^{-1}B$ .

…

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CBSE 2025Set ANNUAL1 markMCQ
$x$
(i) $-2$
(ii) $-1$
(iii) $2$
(iv) $1$
›Reveal solutionSolution

Multiply out the matrices and compare entries.

$\begin{bmatrix}1&3\\2&1\end{bmatrix}\begin{bmatrix}1\\x\end{bmatrix}=\begin{bmatrix}1(1)+3(x)\\2(1)+1(x)\end{bmatrix}=\begin{bmatrix}1+3x\\2+x\end{bmatrix}$

Setting this equal to $\begin{bmatrix}7\\4\end{bmatrix}$ :

…

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CBSE 2025Set ANNUAL1 markMCQ
$x$
(a) $x = -\dfrac{1}{3}, y = 7$
(b) Not possible to find
(c) $x = -\dfrac{2}{3}, y = 7$
(d) $x = -\dfrac{1}{3}, y = -\dfrac{2}{3}$
›Reveal solutionSolution

Equating corresponding entries gives two different equations for x that contradict each other, so no consistent solution exists.

For $\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix} = \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$ , equating each entry:

$3x+7 = 0 \Rightarrow x = -\dfrac{7}{3}$

$5 = y-2 \Rightarrow y = 7$

$y+1 = 8 \Rightarrow y = 7$ (consistent with above)

$2-3x = 4 \Rightarrow x = -\dfrac{2}{3}$

…

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CBSE 2025Set ANNUAL1 markMCQ
$A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$
(a) $19$
(b) $\dfrac{1}{19}$
(c) $-19$
(d) $-\dfrac{1}{19}$
›Reveal solutionSolution

Compute |A| and adj(A), form A⁻¹, and compare it entry-by-entry with kA.

$A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ , $|A| = 2(-2) - 3(5) = -4-15 = -19$

$\text{adj}(A) = \begin{bmatrix} -2 & -3 \\ -5 & 2 \end{bmatrix}$

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. If [[x-2y, 0], [5, x]] = [[-5, 0], [5, 3]], then y is equal to:$
(a) 1
(b) 3
(c) 2
(d) 4
›Reveal solutionSolution

Equal matrices have equal corresponding entries — match the (2,2) entries first to get $x$ , then use the (1,1) entry to get $y$ .

Given $\begin{pmatrix} x-2y & 0 \\ 5 & x\end{pmatrix} = \begin{pmatrix} -5 & 0 \\ 5 & 3\end{pmatrix}$ .

Comparing the (2,2) entries: $x = 3$ .

…

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CBSE 2025Set ANNUAL1 mark
$Q. If [[a+4, 3b], [8, -14]] = [[2a+2, b+4], [8, a-8b]], then find the value of a + b.$
›Reveal solutionSolution

Equate corresponding entries of the two equal matrices to get $a=2$ , $b=2$ , so $a+b=4$ .

Two matrices are equal only if every corresponding entry is equal. Comparing entries of

$\begin{bmatrix} a+4 & 3b \\ 8 & -14 \end{bmatrix} = \begin{bmatrix} 2a+2 & b+4 \\ 8 & a-8b \end{bmatrix}$ :

From the (1,1) entries: $a+4 = 2a+2 \Rightarrow 2 = a \Rightarrow a = 2$ .

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. If A = [[2x, 0], [x, x]] and A⁻¹ = [[1, 0], [-1, 2]], then x equals -$
(i) 1
(ii) 2
(iii) 1/2
(iv) −2
›Reveal solutionSolution

Compute $A^{-1}$ from $A=\begin{pmatrix}2x&0\\x&x\end{pmatrix}$ using the $2\times2$ inverse formula and match it to the given $A^{-1}$ .

For $A=\begin{pmatrix}2x&0\\x&x\end{pmatrix}$ , $\det A = (2x)(x)-(0)(x)=2x^2$ .

Using $A^{-1}=\dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ :

$A^{-1}=\dfrac{1}{2x^2}\begin{pmatrix}x&0\\-x&2x\end{pmatrix}=\begin{pmatrix}\tfrac{1}{2x}&0\\-\tfrac{1}{2x}&\tfrac{1}{x}\end{pmatrix}$ .

…

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CBSE 20241 markMCQ
$\begin{bmatrix} 8 & 14 \\ 9 & 7 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} X$
›Reveal solutionSolution
We solve the matrix equation $A = BX$ by left-multiplying both sides by $B^{-1}$ , giving $X = B^{-1}A$ . Computing the inverse of $B = \begin{bmatrix}1&2\\3&1\end{bmatrix}$ and multiplying yields $X = \begin{bmatrix}2&0\\3&7\end{bmatrix}$ , which matches option (C).

The core idea here is that a matrix equation like $A = BX$ is solved exactly like the scalar equation $a = bx$ — you isolate $X$ by multiplying both sides by the inverse of $B$ . But because matrix multiplication is not commutative, you must multiply on the left by $B^{-1}$ , not on the right. That single detail is the entire key.

Let’s walk through it.

Set up the equation clearly. We are given

$\begin{bmatrix} 8 & 14 \\ 9 & 7 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} X.$
Call the left matrix $A$ and the coefficient matrix $B$ , so $A = BX$ . Our job is to find $X$ .

Why left-multiplication by $B^{-1}$ works. If $B$ is invertible, then $B^{-1}B = I$ , the identity matrix. Multiplying both sides of $A = BX$ on the left by $B^{-1}$ gives

$B^{-1}A = B^{-1}(BX) = (B^{-1}B)X = IX = X.$
So $X = B^{-1}A$ . Notice: if we had multiplied on the right instead, we’d get $AB^{-1}$ , which is a completely different (and wrong) matrix.

Watch out

A common mistake is to write $X = A B^{-1}$ by analogy with scalars. But matrix multiplication is not commutative — $B^{-1}A \neq A B^{-1}$ in general. Always multiply on the side where the inverse cancels the original matrix.

Find $B^{-1}$ . For a $2 \times 2$ matrix $B = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the inverse is

$B^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix},$
provided the determinant $ad - bc \neq 0$ .

Here $a=1$ , $b=2$ , $c=3$ , $d=1$ . The determinant is
$\det(B) = (1)(1) - (2)(3) = 1 - 6 = -5.$
So
$B^{-1} = \frac{1}{-5} \begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -\frac{1}{5} & \frac{2}{5} \\ \frac{3}{5} & -\frac{1}{5} \end{bmatrix}.$

Multiply $B^{-1}A$ . Now $A = \begin{bmatrix} 8 & 14 \\ 9 & 7 \end{bmatrix}$ . Compute $X = B^{-1}A$ :

$X = \begin{bmatrix} -\frac{1}{5} & \frac{2}{5} \\ \frac{3}{5} & -\frac{1}{5} \end{bmatrix} \begin{bmatrix} 8 & 14 \\ 9 & 7 \end{bmatrix}.$
Multiply entry by entry: …
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Part (a)

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