Matrices

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.

Step 2: Equate corresponding elements.
Given: So, \begin{align*} -5 &= 9y \\ 2 &= 6z \\ 6 &= 2x \\ 3 &= 3 \end{align*}
Step 3: Solve for .
Conclusion:
The required value of is:

Concept: Three points are collinear if the area of the triangle formed by them is zero. Using determinant method: Step 1: Substitute the given points.

Step 2: Expand the determinant.

Step 3: Set equal to zero.
Correction:
Rechecking calculation carefully: Conclusion:
The correct value of is:

Concept: A square matrix is said to be skew-symmetric if: Key properties of skew-symmetric matrices: \begin{itemize} \item All diagonal elements are zero. \item The determinant of a skew-symmetric matrix of odd order is always zero. \end{itemize} Step 1: Use determinant property.
For any square matrix,
Step 2: Apply skew-symmetric condition.
Since , taking determinants:
Step 3: Use determinant rule.
For an matrix: Thus,
Step 4: Consider odd order.
If is odd, then . So, Conclusion:
The determinant of a skew-symmetric matrix of odd order is always zero.