Question

Find a matrix X such that 2A + B + X = 0 , where

\(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\rm B =\ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ ?\)

1. \(\begin{bmatrix} 1 & 2 \\\ 7 & 13 \end{bmatrix}\)
2. \(\begin{bmatrix} -1 & -2 \\\ -7 & -13 \end{bmatrix}\)
3. \(\begin{bmatrix} 13 & 2 \\\ 7 & 1 \end{bmatrix}\)
4. \(\begin{bmatrix} -13 & -2 \\\ -7 & -1 \end{bmatrix}\)

Answer 1

Correct Answer - Option 2 : \(\begin{bmatrix} -1 & -2 \\\ -7 & -13 \end{bmatrix}\)

Concept:

Two matrices may be added or subtracted only if they have the same dimension; that is, they must have the same number of rows and columns. 

Addition or subtraction is accomplished by adding or subtracting corresponding elements.

 

Calculations:

Given, \(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\; \rm B = \ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ \)

Consider, 2A + B + X = 0

⇒ \(\rm 2\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} + \ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ \) + X = \(\begin{bmatrix} 0 & 0\\ 0 &0 \end{bmatrix}\)

Two matrices may be added or subtracted only if they have the same dimension; that is, they must have the same number of rows and columns. Addition or subtraction is accomplished by adding or subtracting corresponding elements.

⇒ \(\rm \begin{bmatrix} 1 & 2 \\\ 7 & 13 \end{bmatrix} \ \) + X = \(\begin{bmatrix} 0 & 0\\ 0 &0 \end{bmatrix}\)

⇒ X = \(\begin{bmatrix} 0 & 0\\ 0 &0 \end{bmatrix}\) - \(\rm \begin{bmatrix} 1 & 2 \\\ 7 & 13 \end{bmatrix} \ \)

⇒ X = \(\rm \begin{bmatrix} -1 & -2 \\\ - 7 & -13 \end{bmatrix} \ \)