Question

(1) Verify by direct multiplication that the given matrices are inverses of one another. A = [4 9; 3 7], B = [7 -9; -3 4] (2) Let A = [-1 -2 3; -1 1 1; -1 -2 -1]. Find the third column vector of A^(-1) without determining the other columns of the inverse matrix. (use the set up from the video before the Gauss-Jordan algorithm was introduced). (3) Determine A^(-1), if possible, using the algorithm introduced in the video lesson. If A^(-1) exists, check your answer by verifying that AA^(-1) = I_n. (a) A = [1 -1 2; 2 1 11; 4 -3 10] (b) A = [0 1 0; 0 0 1; 0 1 2] (4) Use A^(-1) to find the solution to the given system. x1 + x2 - 2x3 = -2 x2 + x3 = 3 2x1 + 4x2 - 3x3 = 1 (5) Determine if the given statement is true or false. Justify (if false, use a counter example). (a) Every square matrix that does not contain a row of zeros is invertible. (b) If A and B are invertible matrices, then so is AB. (c) If A is an invertible matrix such that A^2 = A, then A is the identity matrix.

          (1) Verify by direct multiplication that the given matrices are inverses of one another.

A = [4 9; 3 7], B = [7 -9; -3 4]

(2) Let A = [-1 -2 3; -1 1 1; -1 -2 -1]. Find the third column vector of A^(-1) without determining the other columns of the inverse matrix. (use the set up from the video before the Gauss-Jordan algorithm was introduced).

(3) Determine A^(-1), if possible, using the algorithm introduced in the video lesson. If A^(-1) exists, check your answer by verifying that AA^(-1) = I_n.

(a) A = [1 -1 2; 2 1 11; 4 -3 10]

(b) A = [0 1 0; 0 0 1; 0 1 2]

(4) Use A^(-1) to find the solution to the given system.
x1 + x2 - 2x3 = -2
x2 + x3 = 3
2x1 + 4x2 - 3x3 = 1

(5) Determine if the given statement is true or false. Justify (if false, use a counter example).

(a) Every square matrix that does not contain a row of zeros is invertible.

(b) If A and B are invertible matrices, then so is AB.

(c) If A is an invertible matrix such that A^2 = A, then A is the identity matrix.

(1) Verify by direct multiplication that the given matrices are inverses of one another.

A = [4 9; 3 7], B = [7 -9; -3 4]

(2) Let A = [-1 -2 3; -1 1 1; -1 -2 -1]. Find the third column vector of A^(-1) without determining the other columns of the inverse matrix. (use the set up from the video before the Gauss-Jordan algorithm was introduced).

(3) Determine A^(-1), if possible, using the algorithm introduced in the video lesson. If A^(-1) exists, check your answer by verifying that AA^(-1) = In.

(a) A = [1 -1 2; 2 1 11; 4 -3 10]

(b) A = [0 1 0; 0 0 1; 0 1 2]

(4) Use A^(-1) to find the solution to the given system.
x1 + x2 - 2x3 = -2
x2 + x3 = 3
2x1 + 4x2 - 3x3 = 1

(5) Determine if the given statement is true or false. Justify (if false, use a counter example).

(a) Every square matrix that does not contain a row of zeros is invertible.

(b) If A and B are invertible matrices, then so is AB.

(c) If A is an invertible matrix such that A^2 = A, then A is the identity matrix.

Added by Sydney K.

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