Linear Algebra

Concept: In matrix algebra, the transpose of a matrix is obtained by interchanging its rows and columns. If is an matrix, then its transpose is an matrix. One important property of matrix transpose is related to the product of two matrices.
Step 1: Recall the transpose property of matrix multiplication.
For any two matrices and whose product is defined, the transpose of their product satisfies: This means the order of multiplication reverses when taking the transpose.
Step 2: Understand the reversal of order.
Matrix multiplication is not commutative, meaning: Therefore, when taking the transpose of a product, the matrices must appear in reverse order.
Step 3: Conclusion.
Using the transpose rule for matrix multiplication: Hence, the correct answer is:

Concept: An identity matrix of order is a square matrix in which:

• All diagonal elements are .
• All non-diagonal elements are .

The general form of an identity matrix is: A key property of determinants is that the determinant of a triangular matrix (including diagonal matrices) equals the product of its diagonal elements.
Step 1: Identify the diagonal elements.
For an identity matrix , all diagonal elements are equal to .
Step 2: Apply the determinant rule.
The determinant of a diagonal matrix is the product of its diagonal elements.
Step 3: Compute the result.
Since the product of ones is:
Step 4: Conclusion.
Therefore, the determinant of an identity matrix of any order is: