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CBSE Board Examination 2023 · Section A · 1 mark

Q4: B when 2A+B2A + B is a null matrix

Board 2023Ch 3 · Matrices
The question

If A=[3452]A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix} and 2A+B2A + B is a null matrix, then B is equal to :

(a) [68104]\begin{bmatrix} 6 & 8 \\ 10 & 4 \end{bmatrix}

(b) [−6−8−10−4]\begin{bmatrix} -6 & -8 \\ -10 & -4 \end{bmatrix}

(c) [58103]\begin{bmatrix} 5 & 8 \\ 10 & 3 \end{bmatrix}

(d) [−5−8−10−3]\begin{bmatrix} -5 & -8 \\ -10 & -3 \end{bmatrix}

Exam answer

2A+B=O⇒B=−2A2A + B = O \Rightarrow B = -2A⁠.

Why?

A null matrix OO has every entry 0. Subtract 2A2A from both sides, just as with numbers:

B=O−2A=−2A\begin{aligned} B &= O - 2A = -2A \end{aligned}
B=−2[3452]=[−2(3)−2(4)−2(5)−2(2)]=[−6−8−10−4]\begin{aligned} B &= -2\begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix} \\ &= \begin{bmatrix} -2(3) & -2(4) \\ -2(5) & -2(2) \end{bmatrix} \\ &= \begin{bmatrix} -6 & -8 \\ -10 & -4 \end{bmatrix} \end{aligned}
Why?

A scalar multiple multiplies every entry by the scalar, here −2-2⁠.

Check: 2A2A has entries 6,8,10,46, 8, 10, 4⁠, and adding −6,−8,−10,−4-6, -8, -10, -4 gives 0 in every place.

Watch out: option (a) is 2A2A itself, with the sign lost.

Answer: (b) [−6−8−10−4]\begin{bmatrix} -6 & -8 \\ -10 & -4 \end{bmatrix}

Where this comes from

Chapter 3, Matrices: Addition and scalar multiples of matrices.

Question type: Find the unknown matrix XX

How to spot it: click to see

Type 4. Find the unknown matrix XX

Spot it: a matrix equation with an unknown matrix XX in it, such as 2X+Y2X + Y equal to a given matrix.

Here: 2A+B=O2A + B = O has the unknown matrix BB in it: take 2A2A across to get B=−2AB = -2A⁠, then multiply every entry of AA by −2-2⁠.

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