If a is an invertible square matrix with determinant d (not equal to 0) the determinant of a⁻¹ and at is |A|⁻¹.
Invertible square matrix
An invertible square matrix refers the square matrix that has an inverse. We say that a square matrix is invertible if and only if the determinant is not equal to zero.
Given,
Here we need to identify the type of matrix, if a is an invertible square matrix with determinant d (not equal to 0) the determinant of a⁻¹.
As we know that,
Here we have the invertible square matrix with the determinant d.
And the determinant of A⁻¹ is written as,
As per the concept of matrix,
AA⁻¹ = I
When we apply the Det on both sides then we get,
|AA⁻¹| = |I|
We know that the value of |I| = 1, so,
|AA⁻¹| = 1
So, it can be written as,
|A⁻¹| = 1/ |A|
Therefore,
|A⁻¹| = |A⁻¹|
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