A. If A and B are two invertible matrix, then (AB)-1=A-1 B-1
B. Every skew- symmetric matrix of odd order is invertible
C. If A is non- singular matrix, then(𝐴𝑇)−1=(𝐴−1)𝑇
D. If A is an involutory matrix, then(I+A)(I-A)=0
E. A diagonal matrix is both an upper triangular and a lower triangular
Choose the correct answer from the options given below:
A. If and are two invertible matrices, then
False. The correct property for the inverse of a product is the Reversal Law:
B. Every skew-symmetric matrix of odd order is invertible
False. The determinant of a skew-symmetric matrix of odd order is always zero (). Since the determinant is zero, the matrix is singular and not invertible.
C. If is a non-singular matrix, then
True. The operations of taking the transpose and taking the inverse are commutative.
D. If is an involutory matrix, then
True. An involutory matrix satisfies the condition .
Expanding the product:
Since :
E. A diagonal matrix is both an upper triangular and a lower triangular
True. By definition, an upper triangular matrix has all entries below the diagonal as zero, and a lower triangular matrix has all entries above the diagonal as zero. A diagonal matrix satisfies both conditions.
Based on the evaluation above:
C is correct.
D is correct.
E is correct.
The most appropriate answer from the options (depending on your provided list) would be:
C, D, and E only.
