Question

Which statement about inverse matrices is true

Answer 1

We will verify each options

option-A:

Inverse matrix:

we know that inverse matrix always exists for square matrix and non-singular matrix

non-singular matrix means $|A|\neq 0$

so, this is TRUE

option-B:

Inverse matrix:

we know that inverse matrix always exists for square matrix and non-singular matrix

non-singular matrix means $|A|\neq 0$

so, this is FALSE

option-C:

Inverse matrix:

we know that inverse matrix always exists for square matrix and non-singular matrix

non-singular matrix means $|A|\neq 0$

so, this is FALSE

option-D:

We know property of inverse matrix

$AA^{-1} =I$

where I is the identity matrix

But this is not true for non-square matrix

so, this is FALSE

Answer 2

Answer: Option A.

Step-by-step explanation:

If B is the inverse matrix of the matrix A, we have that:

A*B = B*A = I

This means that both matrixes commute with each other, this implies that the matrixes need to have the same number of rows than columns, so the matrices are square.

Now the other 3 statements are false:

If the determinant is zero the matrix is not invertible.

Not all the square matrices are invertible.

and option B is obviously false because is the negation of option A.