# Determinants and the Inverse of a Matrix

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## Determinants and the Inverse of a Matrix Answer Key

1.

For [math]M = [[6,-3],[-2,1]][/math], find [math]|M|[/math].

- -9
- 16
- 12
- 0

2.

What is the determinant of [math] [[5,7,-2],[3,0,1],[-9,-7,2]] ?[/math]

- -112
- -28
- 98
- -98

3.

The matrix [math][[5,-1],[-10,2]][/math] has an inverse.

- True
- False

4.

The matrix [math][[1,2,0],[3,4,2],[3,1,5]][/math] has an inverse.

- True
- False

5.

If A is a given square matrix, and it is known that there exists a matrix B such that [math]AB=1[/math], which of the following would be the most efficient ways to find the matrix B?

- Find the inverse of A. This is the matrix B.
- Find the transpose of A. This is the matrix B.
- Create a matrix B whose elements are variables. Then, perform matrix multiplication with the matrix A, setting each resulting entry equal to one. Solve this system of equations, which will give the elements of matrix B.
- Multiply both sides of the equation, on the left, by slight variations of the matrix A. When one of these matrices, multiplied by A, becomes the identity matrix, this is the matrix B.

6.

For [math]M = [[4,-5],[-3,6]][/math], find [math]M^{-1}[/math], if it exists.

- Matrix M does not have an inverse.
- [math] [[2/13, 5/39],[1/13, 4/39]] [/math]
- [math] [[2/3, 5/9],[1/3, 4/9]] [/math]
- [math] [[4/9, -5/9],[-1/3, 2/3]] [/math]

7.

Find the inverse of the following matrix, if it exists. [math][[4,-3,8],[1,0,2],[-5,6,4]][/math]

- There is no inverse of this matrix.
- [math] [[-2/7, 10/7, -1/7],[-1/3, 4/3, 0],[1/7, -3/14, 1/14]][/math]
- [math] [[2/7, -10/7, 1/7],[1/3, -4/3, 0],[-1/7, 3/4, -1/4]][/math]
- [math] [[-2/7, -10/7, -1/7],[1/3, 4/3, 0],[1/7, 3/4, 1/4]][/math]

8.

Matrix A is a square matrix and [math]|A| = -3[/math]. With this information, which of the following has to be true about matrix A?

- It is invertible.
- It has more negative elements than positive elements.
- It has an odd number of rows and columns.
- One cannot take the transpose of it.

9.

Sarah has been given the following matrix equation to solve, [math]A x = b[/math], where A is a 3-by-3 matrix, b is a 3-by-1 matrix, and x is a 3-by-1 matrix. If she knows the equation can be solved by multiplying the inverse of A on both sides of the equation, which of the following must be true?

- A must be a diagonal matrix (its off-diagonal entries must be zero).
- The matrix b must also have an inverse.
- The determinant of A is not equal to zero.
- More than half of the elements in the matrix A are non-zero.

10.

James needs to show that for matrix [math]A = [[5,-2,1],[3,3,2],[-6,3,-1]] [/math], there is no matrix [math]B, B!=I[/math], such that [math]AB = I[/math], where [math]I[/math] is the 3-by-3 identity matrix. How can he do this?

- Try at least 3 matrices, and if none of them multiplied by [math]A[/math] equal the identity matrix, then it is not possible.
- Find the inverse of [math]A[/math], and then show that since this matrix is unique, there cannot exist another matrix [math]B[/math] such that [math]AB = I[/math].
- Subtract by the additive inverse on both sides, and then factor the left hand side of the equation. This implies that if [math]B=I[/math] the equation equals the zero matrix, which it can't.
- Show that the determinant of [math]A[/math] is zero, which means that it does not have a multiplicative inverse.

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