Question

Let $A = \begin{bmatrix} 2 & x \\ 3 & 1 \end{bmatrix}, B = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $C = \begin{bmatrix} 3x+2 & 10 \\ 14-x & 10 \end{bmatrix}$. If $B^T A^T = C$, then the value of $x$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

Always use $(AB)^T = B^T A^T$ to avoid transposing multiple matrices separately. Transpose the final result once to save time.

Answer 1

The Correct Option is B

To solve the problem, we need to find the value of \( x \) such that the equation \( B^T A^T = C \) holds true, where matrices \( A, B, \) and \( C \) are given as:

Matrix	Expression
\( A \)	\(\begin{bmatrix} 2 & x \\ 3 & 1 \end{bmatrix}\)
\( B \)	\(\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\)
\( C \)	\(\begin{bmatrix} 3x+2 & 10 \\ 14-x & 10 \end{bmatrix}\)

Step 1: Transpose the matrices \( A \) and \( B \).

The transpose of matrix \( A \), denoted \( A^T \), is:

\(A^T = \begin{bmatrix} 2 & 3 \\ x & 1 \end{bmatrix}\)

The transpose of matrix \( B \), denoted \( B^T \), is:

\(B^T = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}\)

Step 2: Calculate the product \( B^T A^T \).

We multiply the matrices \( B^T \) and \( A^T \) as follows:

\(B^T A^T = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ x & 1 \end{bmatrix}\)

The product is calculated by taking the dot product of rows of \( B^T \) with columns of \( A^T \):

• Row 1, Column 1: \((3 \cdot 2) + (1 \cdot x) = 6 + x\)
• Row 1, Column 2: \((3 \cdot 3) + (1 \cdot 1) = 9 + 1 = 10\)
• Row 2, Column 1: \((2 \cdot 2) + (4 \cdot x) = 4 + 4x\)
• Row 2, Column 2: \((2 \cdot 3) + (4 \cdot 1) = 6 + 4 = 10\)

This gives us the product \( B^T A^T \):

\(B^T A^T = \begin{bmatrix} 6 + x & 10 \\ 4 + 4x & 10 \end{bmatrix}\)

Step 3: Equate \( B^T A^T \) and \( C \) to find \( x \).

We know that \( B^T A^T = C \). Therefore, we set the corresponding elements of the two matrices equal:

• \(6 + x = 3x + 2\)
• \(4 + 4x = 14 - x\)

From the first equation, solve for \( x \):

\(6 + x = 3x + 2 \Rightarrow 6 - 2 = 3x - x \Rightarrow 4 = 2x \Rightarrow x = 2\)

Verify this value with the second equation:

\(4 + 4(2) = 14 - 2 \Rightarrow 4 + 8 = 12, \Rightarrow 12 = 12\)

This confirms the calculations are correct.

The value of \( x \) is \(2\).