Problem 28 - Algebra 2

Question

Perform the indicated matrix operation. If the matrix does not exist, write impossible. $$ 5\left[\begin{array}{r}{1} \\ {-1} \\\ {-3}\end{array}\right]+6\left[\begin{array}{r}{-4} \\ {3} \\\ {5}\end{array}\right]-2\left[\begin{array}{r}{-3} \\ {8} \\\ {-4}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} -13 \\ -3 \\ 23 \end{bmatrix} \).

Step 1: Distribute the Scalars

First, multiply each matrix by its corresponding scalar. This involves multiplying each entry of the matrix by the scalar.1. Multiply matrix \( \begin{bmatrix} 1 \ -1 \ -3 \end{bmatrix} \) by 5: \[ 5 \begin{bmatrix} 1 \ -1 \ -3 \end{bmatrix} = \begin{bmatrix} 5 \ -5 \ -15 \end{bmatrix} \]2. Multiply matrix \( \begin{bmatrix} -4 \ 3 \ 5 \end{bmatrix} \) by 6: \[ 6 \begin{bmatrix} -4 \ 3 \ 5 \end{bmatrix} = \begin{bmatrix} -24 \ 18 \ 30 \end{bmatrix} \]3. Multiply matrix \( \begin{bmatrix} -3 \ 8 \ -4 \end{bmatrix} \) by -2: \[ -2 \begin{bmatrix} -3 \ 8 \ -4 \end{bmatrix} = \begin{bmatrix} 6 \ -16 \ 8 \end{bmatrix} \]

Step 2: Add the Resulting Matrices

Now, add the resulting matrices obtained from Step 1. This involves adding each corresponding entry of the matrices.1. Add the matrices \( \begin{bmatrix} 5 \ -5 \ -15 \end{bmatrix} \), \( \begin{bmatrix} -24 \ 18 \ 30 \end{bmatrix} \), and \( \begin{bmatrix} 6 \ -16 \ 8 \end{bmatrix} \). Perform the following calculations for each entry: - First entry: \( 5 + (-24) + 6 = -13 \) - Second entry: \( -5 + 18 + (-16) = -3 \) - Third entry: \( -15 + 30 + 8 = 23 \)The resulting matrix is:\[\begin{bmatrix}-13 \ -3 \ 23 \end{bmatrix}\]

Step 3: Conclusion

The final matrix, after performing all indicated operations, is \( \begin{bmatrix} -13 \ -3 \ 23 \end{bmatrix} \). This matrix exists, confirming the operation was possible.

Key Concepts

Scalar Multiplication

Scalar multiplication is the process of multiplying each element of a matrix by a scalar value. It's very much like scaling a vector but applied to each individual entry in the matrix. For example, if you are given a matrix \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and you want to multiply it by a scalar 3, you simply multiply each of the elements by 3:

• Top-left: \( 1 \times 3 = 3 \)
• Top-right: \( 2 \times 3 = 6 \)
• Bottom-left: \( 3 \times 3 = 9 \)
• Bottom-right: \( 4 \times 3 = 12 \)

The resulting matrix is then \( \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} \).
When performing scalar multiplication, you always multiply the same scalar by each element in the matrix no matter what its original value was.

Matrix Addition

Matrix addition involves the entry-wise addition of two matrices that are of the same dimensions. This means you can only add matrices that have the same number of rows and columns.
Imagine you have the following matrices:

• Matrix A: \( \begin{bmatrix} 2 & 4 \ 1 & 3 \end{bmatrix} \)
• Matrix B: \( \begin{bmatrix} 5 & 1 \ 7 & 6 \end{bmatrix} \)

To add them, each entry in Matrix A is added to the corresponding entry in Matrix B:

• Top-left: \( 2 + 5 = 7 \)
• Top-right: \( 4 + 1 = 5 \)
• Bottom-left: \( 1 + 7 = 8 \)
• Bottom-right: \( 3 + 6 = 9 \)

Thus, the resulting matrix after addition is \( \begin{bmatrix} 7 & 5 \ 8 & 9 \end{bmatrix} \).
Remember, if the matrices are not the same size, matrix addition is impossible.

Linear Combinations

A linear combination of matrices refers to an expression constructed from a set of matrices that involves scalar multiplication and matrix addition.For instance, consider the equation:\[ c_1\times A_1 + c_2\times A_2 + c_3\times A_3 \]where \( c_1, c_2, \) and \( c_3 \) are scalars and \( A_1, A_2, \) and \( A_3 \) are matrices.

Here's how we apply it:1. **Scalar multiplication:** Each matrix \( A_i \) is multiplied by its corresponding scalar \( c_i \).2. **Matrix addition:** The results from the scalar multiplications are then added together as matrices.
Linear combinations are central to understanding matrix transformations and solving systems of linear equations which are essentially represented in matrix form.

Matrix Arithmetic

Matrix arithmetic encompasses operations such as scalar multiplication, matrix addition, subtraction, and sometimes multiplication. These operations follow specific rules:

• **Scalar Multiplication:** Each entry in the matrix is multiplied by the scalar.
• **Matrix Addition/Subtraction:** Only matrices of the same dimensions can be added/subtracted, and it involves adding/subtracting corresponding entries.
• **Matrix Multiplication:** More complex, involves multiplying rows by columns, only possible when the number of columns in the first matrix is equal to the number of rows in the second.

Matrix arithmetic extends your ability to manipulate and solve equations, especially in fields like engineering and physics, by using matrices to represent and simplify complex systems and relationships.