Inverses: 3x3 Matrices (9.9.4)

Question 1

• A 3x3 matrix is an array with three columns and three rows.

Answer 1

• A 3x3 matrix is an array with three columns and three rows.

Question 2

• An inverse matrix is one that, when multiplied with its original matrix, yields the identity matrix.

Answer 2

• An inverse matrix is one that, when multiplied with its original matrix, yields the identity matrix.

Question 3

• The only matrices that have inverses are those which are square and nonsingular :

Answer 3

• The only matrices that have inverses are those which are square and nonsingular :

Question 4

• To check whether one matrix is the inverse of another, multiply the original matrix with the proposed inverse matrix. If the answer is the identity matrix, then the second matrix is the inverse of the first matrix.

Answer 4

• To check whether one matrix is the inverse of another, multiply the original matrix with the proposed inverse matrix. If the answer is the identity matrix, then the second matrix is the inverse of the first matrix.

Question 5

• To find the inverse of any matrix:
1. Set up a matrix alternating “+” and “–” .
2. Compute the determinant of the matrix.
3. For each position in the new matrix:
(a) block out the element in that position in the original matrix plus everything in both its row and
its column.
(b) Evaluate the determinant of whatever is still uncovered.
(c) Write that value in the new matrix; change its sign if it lies in a “–” position.
4. Flip this new matrix over the main diagonal, which runs from the upper left to the lower right.
5. Multiply this new matrix by the reciprocal of the determinant of the original matrix.

Answer 5

• To find the inverse of any matrix:
1. Set up a matrix alternating “+” and “–” .
2. Compute the determinant of the matrix.
3. For each position in the new matrix:
(a) block out the element in that position in the original matrix plus everything in both its row and
its column.
(b) Evaluate the determinant of whatever is still uncovered.
(c) Write that value in the new matrix; change its sign if it lies in a “–” position.
4. Flip this new matrix over the main diagonal, which runs from the upper left to the lower right.
5. Multiply this new matrix by the reciprocal of the determinant of the original matrix.