Problem 26 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{rr}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(2 C+B\)

Step-by-Step Solution

Answer

The result of \( 2C + B \) is \( \begin{bmatrix} -9 & 24 \\ 10 & 4 \end{bmatrix} \).

Step 1: Determine Matrices Dimensions

Matrix \( C \) is a \( 2 \times 2 \) matrix and matrix \( B \) is also a \( 2 \times 2 \) matrix. To perform addition, both matrices must have the same dimensions. Since both matrices are \( 2 \times 2 \), the operation is possible.

Step 2: Multiply Matrix C by Scalar

Multiply each element of matrix \( C \) by 2. This gives us: \[ 2C = \begin{bmatrix} 0 \times 2 & 9 \times 2 \ 7 \times 2 & 1 \times 2 \end{bmatrix} = \begin{bmatrix} 0 & 18 \ 14 & 2 \end{bmatrix} \]

Step 3: Add Matrices

Add the resulting matrix from \( 2C \) to matrix \( B \) as follows: \[ B = \begin{bmatrix} -9 & 6 \ -4 & 2 \end{bmatrix} \] Add corresponding elements: \[ 2C + B = \begin{bmatrix} 0 + (-9) & 18 + 6 \ 14 + (-4) & 2 + 2 \end{bmatrix} = \begin{bmatrix} -9 & 24 \ 10 & 4 \end{bmatrix} \]

Key Concepts

Matrix Addition

Matrix addition is like adding two sets of numbers that are arranged in a grid. This operation is only possible if the two matrices you want to add have the same dimensions. Essentially, this means they must have the same number of rows and columns. Let's say each matrix is a box, and for these boxes to add up perfectly, they need to be the same shape and size.

To add matrices together, simply add the numbers in the same positions from each matrix.

• The numbers in the top left of each matrix form a pair, the top right numbers form another pair, and this pattern continues throughout.
• You add each pair to get a new matrix.

For example, if you have two matrices, each with two rows and two columns, you add each element from the first matrix with its corresponding element from the second matrix. This process gives you a new matrix of the same size, containing the sums.

In the exercise, we added matrices from the previous step, and obtaining the result was straightforward because the matrices had matching dimensions.

Scalar Multiplication

Scalar multiplication is different from matrix addition and involves multiplying an entire matrix by a single number (a scalar). This process is straightforward: you take that single number and multiply it by each and every number within the matrix.

This is akin to scaling the matrix up or down, just like how you might increase or decrease the volume of a stereo using a volume knob. The number you multiply with is called the "scalar," and every element in the matrix gets multiplied by this scalar. This changes each element to be either larger or smaller based on the scalar's value.

• If you multiply by a number larger than 1, the matrix values get larger.
• If it's between 0 and 1, the values get smaller.
• The matrix values will flip in sign if you use a negative scalar.

In the exercise, multiplying matrix \( C \) by 2 doubled each of its elements. This turned the matrix into a scaled-up version of the original.

Matrix Dimensions

Matrix dimensions tell us the size of the matrix. They are expressed as rows x columns. The term "rows" refers to the horizontal lines of numbers, and "columns" to the vertical lines.

Every matrix is represented by its dimensions; for example, a 2x2 matrix means it has 2 rows and 2 columns. Understanding matrix dimensions is crucial because it guides you in performing operations like addition or multiplication. Only matrices with compatible dimensions can be added together.

• For matrix addition, matrices must have exactly the same dimensions.
• Determine dimensions by counting rows first, then columns.

In the step-by-step solution, matrix \( C \) and matrix \( B \) were both 2x2, making it possible to add them. Understanding dimensions is key to successfully performing matrix operations.