Find the products AB and BA to determine whether B is the multiplicative inverse of A. A=[−2132−12],B=[1234]A = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 5

7장, 문제 5

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A = [\begin{matrix} - 2 & 1 \\ \frac{3}{2} & - \frac{1}{2} \end{matrix}],$

검증된 단계별 안내

Step 1: Write down the matrices A and B clearly. Matrix A is given by $A = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$ and matrix B is $B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

Step 2: To find the product $AB$, multiply matrix A by matrix B. Recall that the element in the $i^{th}$ row and $j^{th}$ column of the product matrix is found by taking the dot product of the $i^{th}$ row of A with the $j^{th}$ column of B. For example, the element in the first row and first column of $AB$ is calculated as $(-2)(1) + (1)(3)$.

Step 3: Similarly, calculate all elements of the product matrix $AB$ by performing the dot products for each position: first row second column, second row first column, and second row second column.

Step 4: Next, find the product $BA$ by multiplying matrix B by matrix A. Use the same method of dot products, but this time with rows of B and columns of A.

Step 5: After finding both products $AB$ and $BA$, compare each product to the identity matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If both $AB = I$ and $BA = I$, then matrix B is the multiplicative inverse of matrix A.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. It is essential to correctly compute the products AB and BA to verify if B is the inverse of A. The operation is not commutative, so AB and BA may differ.

Multiplicative Inverse of a Matrix

A matrix B is the multiplicative inverse of matrix A if both AB and BA equal the identity matrix. This means multiplying A by B returns the identity matrix, which acts like 1 in matrix algebra. Checking both products confirms if B is truly the inverse.

Identity Matrix

The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It serves as the multiplicative identity in matrix operations, meaning any matrix multiplied by the identity matrix remains unchanged. Verifying AB = I and BA = I confirms the inverse relationship.

Find the products AB and BA to determine whether B is the multiplicative inverse of A. A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)