Answer to Question #133727 in Linear Algebra for raymond

Question

Suppose A is a 4 * 4 matrix such that det (A) = 16. Find the value of det[4(A^-1)^T ]

Answer 1

Given that,

Determinant of a 4x4 matrix A, |A| = 16

By using the properties of determinants of matrices we can say that,

Determinant of any matrix = Determinant of its transpose

|X^T| = |X|

Hence in our case we can say that,

|4(A^-1)^T| = |4(A^-1)|

By using the property,

|k*X| = kⁿ * |X| where k = constant & n = order of the square matrix, X

Hence in our case, n = 4

$\therefore$ |4(A^-1)| = 4⁴ * |A^-1|

But using the property,

$|X^{-1}| = \frac {1}{|X|}$

Substituting the values in above equation we get,

4⁴ * |A^-1| = 4⁴ * $\frac {1}{16}$ = 16

Hence,

If $|A|=16,\ \ \ \ |(A^{-1})^{T}|=16$