Perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BDA=[2−1253−1]B=[0−2321−5]C=[123−112−121]D=[−2313−24]A=\begin{bmatrix}2 & -1 & 2\\ 5 & 3 & -1\end{bmatrix}\quad B=\begin{bmatrix}0 & -2\\ 3 & 2\\ 1 & -5\end{bmatrix}C=\begin{bmatrix}1 & 2 & 3\\ -1 & 1 & 2\\ -1 & 2 & 1\end{bmatrix}\quad D=\begin{bmatrix}-2 & 3 & 1\\ 3 & -2 & 4\end{bmatrix}

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 23

7장, 문제 23

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

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Identify the dimensions of matrices B and D. For matrix multiplication BD to be defined, the number of columns in B must equal the number of rows in D.

Write down the dimensions of B as $m \times n$ and D as $p \times q$. Check if $n = p$ to confirm if multiplication BD is possible.

If the multiplication is defined, set up the product matrix BD, which will have dimensions $m \times q$.

Calculate each element of the product matrix BD by taking the dot product of the corresponding row of B and column of D. Specifically, the element in row $i$ and column $j$ of BD is given by $\sum_{k=1}^n B_{ik} \times D_{kj}$.

If the multiplication is not defined because the inner dimensions do not match, state that the operation BD is not defined due to incompatible matrix dimensions.

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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 operation is only defined when the number of columns in the first matrix equals the number of rows in the second matrix.

Dimensions of Matrices

The dimensions of a matrix are given as rows × columns. Understanding matrix dimensions is crucial to determine if operations like multiplication are possible and to predict the size of the resulting matrix.

Undefined Matrix Operations

Some matrix operations are undefined if dimension requirements are not met. For example, if the number of columns in the first matrix does not equal the number of rows in the second, multiplication cannot be performed.