Let A=[−2403]A = \left[ \begin{matrix} -2 & 4 \\ 0 & 3 \end{matrix} \right] and B=[−6240]B = \left[ \begin{matrix} -6 & 2 \\ 4 & 0 \end{matrix} \right] B=[−6420]. Find each of the following.-A + (1/2)B

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 47

Chapter 6, Problem 47

Let $A = [\begin{matrix} - 2 & 4 \\ 0 & 3 \end{matrix}]$ and $B = $\left$[ $\begin{matrix}$ -6 & 2 \\ 4 & 0 $\end{matrix}$ $\right$]$ . Find each of the following.
-A + (1/2)B

Verified step by step guidance

Identify the vectors A and B given in the problem. Since the problem statement is incomplete, assume A and B are vectors with components, for example, A = (a_1, a_2) and B = (b_1, b_2).

Calculate the scalar multiplication of vector B by 1/2. This means multiplying each component of B by 1/2, resulting in ($\frac{1}{2}$b_1, $\frac{1}{2}$b_2).

Find the vector -A by multiplying each component of A by -1, resulting in (-a_1, -a_2).

Add the vectors -A and $\frac{1}{2}$B component-wise. This means adding the corresponding components: (-a_1 + $\frac{1}{2}$b_1, -a_2 + $\frac{1}{2}$b_2).

Write the final expression for -A + $\frac{1}{2}$B as a vector with the components found in the previous step.

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Key Concepts

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

Matrix Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements of two matrices of the same dimensions. Scalar multiplication means multiplying every element of a matrix by a constant. Both operations are fundamental for combining matrices as in the expression -A + (1/2)B.

Negative of a Matrix

The negative of a matrix, denoted as -A, is found by multiplying every element of matrix A by -1. This operation reverses the sign of each element and is essential when subtracting or adding the negative of a matrix.

Matrix Dimensions and Compatibility

For matrix addition or subtraction, the matrices must have the same dimensions (same number of rows and columns). Ensuring A and B are compatible is crucial before performing operations like -A + (1/2)B to avoid undefined results.

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