Exam 5: Determinants

A square matrix is called nilpotent if for some positive integer k. Prove that if is nilpotent, then .

Use determinants to find the area of the parallelogram with vertices (0, 0), (3, 2), (2, 3), and (-1, 1).

Without solving for , use Cramer's Rule to find the value of in the linear system.

Find and for the matrix

If A is an invertible matrix, then is invertible, and .

Compute the determinant of the given matrix A by using row operations to reduce to echelon form.

Let A be a matrix. Then .

Find and for the matrix

Evaluate , , , and where ​

For all matrices A, .

If A, B, C, and D are square matrices of the same size, then .

If , then at least one entry of A has value .

Find the area of T (D) where , D is the parallelogram with vertices (4, 2), (5, 3), (4, 6), and (3, 5), and

Use determinants to decide if a unique solution exists for the linear system

Suppose the matrix results from a matrix A using the row operations and . Find .

Compute the determinant of the given matrix A by using row operations to reduce to echelon form.

Without solving for , use Cramer's Rule to find the value of in the linear system.

Use the Shortcut Method to find the determinant of the matrix

If applicable, use Cramer's Rule to find the unique solution of the given linear system. If Cramer's Rule is not applicable, explain why.

If , then .