Sample 2026-27 Class 12 Maths Q6 Solution: Find x when a product of three matrices is O

The question

If $\begin{bmatrix} 1 & x \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix} 4 \\ -1 \end{bmatrix} = O$ ⁠, then value of $x$ is

(A) $-2$

(B) $-1$

(C) $1$

(D) $2$

Exam answer

Multiplying the first two matrices ( $1 \times 2$ by $2 \times 2$ ⁠),

\begin{aligned} &\begin{bmatrix} 1 & x \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \\ &= \begin{bmatrix} 1(1) + x(0) & 1(0) + x(2) \end{bmatrix} \\ &= \begin{bmatrix} 1 & 2x \end{bmatrix} \end{aligned}

Why?

Each entry of a product is a row of the first matrix times a column of the second: multiply matching entries and add.

The row $[1\ \ x]$ times the first column $(1, 0)$ gives $1$ ⁠. The same row times the second column $(0, 2)$ gives $2x$ ⁠. A $1 \times 2$ matrix times a $2 \times 2$ matrix is $1 \times 2$ ⁠.

Then ( $1 \times 2$ by $2 \times 1$ ⁠),

\begin{aligned} &\begin{bmatrix} 1 & 2x \end{bmatrix}\begin{bmatrix} 4 \\ -1 \end{bmatrix} \\ &= [1(4) + 2x(-1)] = [4 - 2x] \end{aligned}

\therefore\ 4 - 2x = 0 \Rightarrow x = 2

Why?

The product is a $1 \times 1$ matrix, so $O$ here is the $1 \times 1$ zero matrix $[0]$ ⁠. Two matrices are equal when their entries are equal, so $4 - 2x = 0$ ⁠.

\begin{aligned} 1(4) &= 4,\quad 2x(-1) = -2x \\[4pt] 4 - 2x &= 0 \\ 4 &= 2x \\ x &= \frac{4}{2} = 2 \end{aligned}

Check: with $x = 2$ ⁠, $[1\ \ 4]$ times the column $(4, -1)$ gives $4 - 4 = 0$ ⁠.

Answer: (D) $2$

Where this comes from

Chapter 3, Matrices: Multiplication of matrices.

Question type: Multiply two matrices

How to spot it: click to see

Type 5. Multiply two matrices: row times column

Spot it: two matrices written side by side, or $AB$ ⁠, $BA$ or $A^2$ ⁠.

Here: Three matrices side by side: multiply two at a time, row times column. The first two give the row $1, 2x$ ⁠, and that times the column $4, -1$ gives $4 - 2x = 0$ ⁠, so $x = 2$ ⁠.

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Q6: Find xxx when a product of three matrices is OOO

Q6: Find $x$ when a product of three matrices is $O$