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} BC + CB

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 39

7장, 문제 39

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.
$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
BC + CB

검증된 단계별 안내

First, identify the matrices A, B, and C from the problem. Here, matrix B is given as $B = \begin{bmatrix}4 & 0 & 5 \\ 1 & 1 & -1 \\ -3 & 5 & 0\end{bmatrix}$ and matrix C is $C = \begin{bmatrix}1 & -2 & -2 \\ -1 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$.

Next, understand that the operation to perform is $BC + CB$, which means you need to multiply matrix B by matrix C, then multiply matrix C by matrix B, and finally add the two resulting matrices.

To multiply two matrices, recall that the number of columns in the first matrix must equal the number of rows in the second matrix. Since both B and C are 3x3 matrices, both products $BC$ and $CB$ are defined and will result in 3x3 matrices.

Calculate the product $BC$ by taking the dot product of the rows of B with the columns of C. For each element $(i,j)$ in the product matrix, multiply corresponding elements from row $i$ of B and column $j$ of C and sum them up.

Similarly, calculate the product $CB$ by taking the dot product of the rows of C with the columns of B. After finding both $BC$ and $CB$, add the corresponding elements of these two matrices to get the final result for $BC + CB$.

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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 only defined when the number of columns in the first matrix equals the number of rows in the second. This operation is not commutative, meaning AB does not generally equal BA.

Matrix Addition

Matrix addition requires two matrices of the same dimensions. Corresponding elements are added together to form a new matrix. This operation is commutative and associative, making it straightforward to combine matrices element-wise.

Determining if an Operation is Defined

Before performing matrix operations, verify dimension compatibility. For multiplication, check if the inner dimensions match; for addition, ensure matrices have identical sizes. If these conditions are not met, the operation is undefined and cannot be performed.