In Exercises 37 - 44, 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} A(BC)

Ch. 6 - Matrices and Determinants

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Ch. 6 - Matrices and Determinants

문제 43

7장, 문제 43

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.
$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
A(BC)

검증된 단계별 안내

Step 1: Identify the matrices A, B, and C as given: $A = \begin{bmatrix} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 1 \\ -2 & -2 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$

Step 2: Determine the dimensions of each matrix to check if the operations are defined: - Matrix A is 3x2 (3 rows, 2 columns) - Matrix B is 2x2 - Matrix C is 2x2

Step 3: Compute the product BC first since the expression is A(BC). Since B is 2x2 and C is 2x2, the product BC is defined and will result in a 2x2 matrix. Use matrix multiplication rules: $ (BC)_{ij} = \sum_{k=1}^{2} B_{ik} \times C_{kj} $

Step 4: After finding BC (a 2x2 matrix), multiply A (3x2) by BC (2x2). Since the number of columns in A (2) matches the number of rows in BC (2), the product A(BC) is defined and will result in a 3x2 matrix. Use matrix multiplication rules again: $ (A(BC))_{ij} = \sum_{k=1}^{2} A_{ik} \times (BC)_{kj} $

Step 5: Perform the multiplications and additions step-by-step for each element of the resulting matrix to find A(BC). Remember to multiply corresponding elements and sum them for each position in the product matrix.

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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. The number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be defined.

Matrix Dimensions and Compatibility

The dimensions of matrices determine whether operations like multiplication are possible. For matrices A (m×n) and B (p×q), multiplication AB is defined only if n = p, resulting in a matrix of dimension m×q.

Associative Property of Matrix Multiplication

Matrix multiplication is associative, meaning (AB)C = A(BC) when the operations are defined. This property allows grouping matrices differently without changing the product, which is useful for simplifying complex matrix expressions.