Question

Find the value of \( x \) if \( \begin{bmatrix} -5 & 6 \\ 2 & 3 \end{bmatrix}^{T} = \begin{bmatrix} 9y & 6z \\ 2x & 3 \end{bmatrix}. \)

Hint:

When matrices are equal, compare corresponding elements directly. Transpose = rows become columns.

Answer 1

Concept: If two matrices are equal, then their corresponding elements are equal. Also, the transpose of a matrix is obtained by interchanging rows and columns. Step 1: Find the transpose of the given matrix.
\[ \begin{bmatrix} -5 & 6 \\ 2 & 3 \end{bmatrix}^{T} = \begin{bmatrix} -5 & 2 \\ 6 & 3 \end{bmatrix} \]
Step 2: Equate corresponding elements.
Given: \[ \begin{bmatrix} -5 & 2 \\ 6 & 3 \end{bmatrix} = \begin{bmatrix} 9y & 6z \\ 2x & 3 \end{bmatrix} \] So, \begin{align*} -5 &= 9y \\ 2 &= 6z \\ 6 &= 2x \\ 3 &= 3 \end{align*}
Step 3: Solve for \( x \).
\[ 6 = 2x \Rightarrow x = 3 \] Conclusion:
The required value of \( x \) is: \[ x = 3 \]