Find the following matrices: A+BA + B A=[2−41],B=[−53−1]A = \begin{bmatrix} 2 \\ -4 \\ 1 \end{bmatrix} , B = \begin{bmatrix} -5 \\ 3 \\ -1 \end{bmatrix}A=2−41,B=−53−1

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 13a

Chapter 7, Problem 13a

Find the following matrices: $A + B$
$A = $\begin{bmatrix}$2 \\-4 \\1$\end{bmatrix}$, B = $\begin{bmatrix}$-5 \\3 \\-1$\end{bmatrix}$$

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Identify the given matrices: $ A = \begin{bmatrix} 2 \\ -4 \\ 1 \end{bmatrix} $ and $ B = \begin{bmatrix} -5 \\ 3 \\ -1 \end{bmatrix} $.

Calculate the scalar multiplication of matrix $ B $ by 2: multiply each element of $ B $ by 2 to get $ 2B = \begin{bmatrix} 2 \times (-5) \\ 2 \times 3 \\ 2 \times (-1) \end{bmatrix} $.

Calculate the scalar multiplication of matrix $ A $ by 5: multiply each element of $ A $ by 5 to get $ 5A = \begin{bmatrix} 5 \times 2 \\ 5 \times (-4) \\ 5 \times 1 \end{bmatrix} $.

Calculate the expression $ A + 2B - 5A $ by performing matrix addition and subtraction element-wise: $ A + 2B - 5A = (A - 5A) + 2B = (-4A) + 2B $.

Perform the element-wise operations to combine the matrices: multiply $ A $ by -4, then add the resulting matrix to $ 2B $ to get the final matrix.

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Key Concepts

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

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is the sum of elements in the same position from the original matrices. This operation is only defined when both matrices have identical sizes.

Scalar Multiplication of a Matrix

Scalar multiplication means multiplying every element of a matrix by a constant (scalar). This operation scales the matrix by the scalar value, changing the magnitude of each element but not the matrix's dimensions.

Matrix Dimensions and Compatibility

Understanding matrix dimensions is crucial for performing operations like addition and scalar multiplication. Two matrices can be added only if they have the same number of rows and columns. Scalar multiplication can be applied to any matrix regardless of its size.

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Find the following matrices: A+BA + B A=[2−41],B=[−53−1]A = \(\begin{bmatrix}\)2 \\-4 \\1\(\end{bmatrix}\), B = \(\begin{bmatrix}\)-5 \\3 \\-1\(\end{bmatrix}\)A=2−41,B=−53−1

Verified video answer for a similar problem:

Key Concepts

Matrix Addition

Scalar Multiplication of a Matrix

Matrix Dimensions and Compatibility

Find the following matrices: $A + B$
$A = \(\begin{bmatrix}\)2 \\-4 \\1\(\end{bmatrix}\), B = \(\begin{bmatrix}\)-5 \\3 \\-1\(\end{bmatrix}\)$