Skip to main content
Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 37
Chapter 7, Problem 37

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

Verified step by step guidance
1
Step 1: Recall the definition of the multiplicative inverse. For a matrix B to be the multiplicative inverse of a matrix A, their product must equal the identity matrix. Specifically, A × B = I and B × A = I, where I is the identity matrix of the same dimension as A and B.
Step 2: Multiply matrix A by matrix B. Use the rules of matrix multiplication, which involve taking the dot product of the rows of A with the columns of B. Write out the resulting matrix explicitly.
Step 3: Multiply matrix B by matrix A. Again, use the rules of matrix multiplication, taking the dot product of the rows of B with the columns of A. Write out the resulting matrix explicitly.
Step 4: Compare the results of A × B and B × A to the identity matrix. The identity matrix has 1s along the main diagonal (from top-left to bottom-right) and 0s elsewhere. Check if both products are equal to the identity matrix.
Step 5: Conclude whether B is the multiplicative inverse of A. If both A × B = I and B × A = I, then B is the multiplicative inverse of A. If not, B is not the multiplicative inverse of A.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
12m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix's elements are calculated by taking the dot product of the rows of the first matrix with the columns of the second matrix. Understanding this operation is crucial for finding the product of matrices A and B.
Recommended video:
03:42
Finding Zeros & Their Multiplicity

Multiplicative Inverse

The multiplicative inverse of a matrix A is another matrix, denoted as A⁻¹, such that when A is multiplied by A⁻¹, the result is the identity matrix I. The identity matrix acts like the number 1 in matrix operations, meaning that A * A⁻¹ = I. To determine if B is the multiplicative inverse of A, one must verify if the product AB equals the identity matrix.
Recommended video:
03:42
Finding Zeros & Their Multiplicity

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning that any matrix multiplied by the identity matrix remains unchanged. For a matrix A of size n x n, the identity matrix I will also be of size n x n, and confirming that the product of A and its supposed inverse B yields I is essential for validating B as A's inverse.
Recommended video:
Guided course
4:35
Introduction to Matrices
Related Practice
Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{3w−4x+y+z=9w+x−y−z=02w+x+4y−2z=3−w+2x+y−3z=3\(\begin{cases}\)3w - 4x + y + z = 9 \(\w\) + x - y - z = 0 \\2w + x + 4y - 2z = 3 \\-w + 2x + y - 3z = 3\(\end{cases}\)

Textbook Question

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

4B - 3C

Textbook Question

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{2x+6y+6z=82x+7y+6z=102x+7y+7z=9The inverse of [266276277] is [720−3−1000−11].\(\begin{cases}\)2x + 6y + 6z = 8 \\2x + 7y + 6z = 10 \\2x + 7y + 7z = 9\(\end{cases}\[\text{The inverse of }\]\begin{bmatrix}\)2 & 6 & 6 \\2 & 7 & 6 \\2 & 7 & 7\(\end{bmatrix}\[\text{ is }\]\begin{bmatrix}\[\frac{7}{2}\) & 0 & -3 \\-1 & 0 & 0 \\0 & -1 & 1\(\end{bmatrix}\]\text{.}\)

Textbook Question

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣0.5750.5390.513∣\(\begin{vmatrix}\)0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\(\end{vmatrix}\)

1
views
Textbook Question

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=02x−y+z=−1−x+3y−z=−8\(\begin{cases}\)x + y + z = 0 \\2x - y + z = -1 \\-x + 3y - z = -8\(\end{cases}\)

Textbook Question

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5