Find the following matrices: a . A+B b. A-B c. -4 A d. 3 A+2 B A=[ 2 -10 -2 14

step 1 For this exercise, we have A plus B, which is going to be 8, 0, negative 4, 14, 0, 6, negative 1

Find the following matrices: a . A+B b. A-B c. -4 A d. 3 A+2...

Find the following matrices:
a . A+B$
b. A-B$
c. -4 A$
d. $3 A+2 B$
$$
A=\left[\begin{array}{rrr}
2 & -10 & -2 \\
14 & 12 & 10 \\
4 & -2 & 2
\end{array}\right], \quad B=\left[\begin{array}{rrr}
6 & 10 & -2 \\
0 & -12 & -4 \\
-5 & 2 & -2
\end{array}\right]
$$

Key Concepts

Linear Combination of Matrices

A linear combination of matrices involves adding together matrices that have each been multiplied by a scalar. For example, an expression like 3A + 2B is a linear combination where matrix A is multiplied by 3 and matrix B is multiplied by 2, with the results then added together element-wise. This operation is fundamental in linear algebra as it forms the basis for many more complex operations and analyses.

Scalar Multiplication

Scalar multiplication is the process of multiplying every entry in a matrix by a constant. This operation scales the entire matrix by the given factor, affecting the magnitude of each element while preserving the overall structure of the matrix.

Matrix Addition

Matrix addition involves adding two matrices by combining their corresponding entries. Each element in the resulting matrix is the sum of the elements at the same position in the original matrices. This operation requires that the matrices have the same dimensions, ensuring that each entry in one matrix has a corresponding entry in the other.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition; however, instead of adding corresponding entries, you subtract them. Each element in the resulting matrix is the difference between the corresponding elements of the two matrices. Like addition, subtraction requires both matrices to have the same dimensions.

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