Question

Let $A=\begin{pmatrix}0& 2\\ 3& 4\end{pmatrix}$ and $I=\begin{pmatrix}1& 0\\ 0& 1\end{pmatrix}$. If $(I+A)\begin{pmatrix}4&-3\\ 2&-1\end{pmatrix}=\begin{pmatrix}8&-5\\ 22& x\end{pmatrix}$, then the value of $x$ is equal to ________.

• A. 14
• B. -14
• C. 12
• D. -12
• E. 15

Hint:

To find $x$, only calculate the entry in the second row and second column.

Answer 1

The Correct Option is B

To solve the problem, we need to find the value of \(x\) in the given matrix equation:

\((I + A)\begin{pmatrix}4 & -3 \\ 2 & -1\end{pmatrix} = \begin{pmatrix}8 & -5 \\ 22 & x\end{pmatrix}\)

Let's go through each step:

Identify the matrices involved: \(I = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\)and \(A = \begin{pmatrix}0 & 2 \\ 3 & 4\end{pmatrix}\)

We need to calculate \(I + A\): \(I + A = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} + \begin{pmatrix}0 & 2 \\ 3 & 4\end{pmatrix} = \begin{pmatrix}1+0 & 0+2 \\ 0+3 & 1+4\end{pmatrix} = \begin{pmatrix}1 & 2 \\ 3 & 5\end{pmatrix}\)

Now, calculate the multiplication: \(\begin{pmatrix}1 & 2 \\ 3 & 5\end{pmatrix} \begin{pmatrix}4 & -3 \\ 2 & -1\end{pmatrix}\)

The resultant product matrix is computed as follows:

First row, first column: \((1 \cdot 4) + (2 \cdot 2) = 4 + 4 = 8\)

First row, second column: \((1 \cdot -3) + (2 \cdot -1) = -3 - 2 = -5\)

Second row, first column: \((3 \cdot 4) + (5 \cdot 2) = 12 + 10 = 22\)

Second row, second column: \((3 \cdot -3) + (5 \cdot -1) = -9 - 5 = -14\)

Compare with the given result matrix: \(\begin{pmatrix}8 & -5 \\ 22 & x\end{pmatrix}\)

From the comparison, the value of \(x\) is \(-14\).

Thus, the value of \(x\) is -14.