Question

4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi = not? Is {ui, } a basis for R2? Why or why

Answer 1

Answer to the above questions :

4.a

Given det(AB) = 20 and det(A) = 4

Since A and B both are 4 x 4 matrices and A is invertible matrix ( because its determinant is non-zero).

Hence, det(AB) = det(A) * det(B)

Therefore, det(B) = 20/4 = 5

(b).

Yes, B is invertible metrix because its determinant is non-zero.

(c).

We know that,

The determinant of a square matrix is the same as the determinant of its transpose.

Hence, det(A^{​​​​​​T}) = det(A) = 4

(d).

We know that,

The determinant of the inverse of an invertible matrix is the inverse of the determinant:

det(A^{​​​​​​-1}) = 1 / det(A) = 1/4 = 0.25

------------------------------------------------