Problem 32 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$4\left[\begin{array}{rr} 1 & -4 \\ 2 & -3 \\ -8 & 4 \end{array}\right]-3\left[\begin{array}{rr} -6 & 9 \\ -2 & 5 \\ -7 & -12 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 22 & -43 \\ 14 & -27 \\ -11 & 52 \end{bmatrix} \).

Step 1: Scalar Multiplication of the First Matrix

Multiply each element of the first matrix by 4. \[4 \begin{bmatrix} 1 & -4 \2 & -3 \-8 & 4\end{bmatrix} = \begin{bmatrix} 4 \times 1 & 4 \times -4 \4 \times 2 & 4 \times -3 \4 \times -8 & 4 \times 4\end{bmatrix} = \begin{bmatrix} 4 & -16 \8 & -12 \-32 & 16\end{bmatrix}\].

Step 2: Scalar Multiplication of the Second Matrix

Multiply each element of the second matrix by 3.\[3 \begin{bmatrix} -6 & 9 \-2 & 5 \-7 & -12\end{bmatrix} = \begin{bmatrix} 3 \times -6 & 3 \times 9 \3 \times -2 & 3 \times 5 \3 \times -7 & 3 \times -12\end{bmatrix} = \begin{bmatrix} -18 & 27 \-6 & 15 \-21 & -36\end{bmatrix}\].

Step 3: Matrix Subtraction

Subtract the second matrix from the first matrix element-wise. \[\begin{bmatrix} 4 & -16 \8 & -12 \-32 & 16\end{bmatrix} - \begin{bmatrix} -18 & 27 \-6 & 15 \-21 & -36\end{bmatrix} = \begin{bmatrix} 4 - (-18) & -16 - 27 \8 - (-6) & -12 - 15 \-32 - (-21) & 16 - (-36)\end{bmatrix}\].

Step 4: Simplify Subtracted Matrix

Simplify each element of the resulting matrix. \[\begin{bmatrix} 4 + 18 & -16 - 27 \8 + 6 & -12 - 15 \-32 + 21 & 16 + 36\end{bmatrix} = \begin{bmatrix} 22 & -43 \14 & -27 \-11 & 52\end{bmatrix}\].

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential concept in matrix operations that makes working with matrices flexible and powerful. Imagine you have a matrix, a grid of numbers. Scalar multiplication involves multiplying every number (element) in that grid by the same single number, known as a scalar.
In terms of the problem given, the scalar multiplication was performed on two matrices using different scalars. Specifically, the first matrix was multiplied by 4, and the second matrix by 3.

• To perform this, simply take each element of the matrix and multiply it by the scalar. For example, if the element is 2 and the scalar is 4, the result will be 8.
• Repeat this process for each element in the matrix.

By multiplying each element individually by the scalar, you get a new matrix, which has the same number of rows and columns as the original. It's like scaling up or down the entire matrix by the factor of the scalar you used. This step helps in balancing and comparing matrices quantitatively.

Matrix Subtraction

Matrix subtraction is a straightforward process, but it's crucial to note that it can only be performed on matrices of the same dimension.
This means both matrices must have the same number of rows and columns.
Matrix subtraction involves taking two matrices and subtracting each element in one matrix from the corresponding element in the other. Here's how it works in our exercise:

• First, align the matrices element by element. Start with the first rows and move across till you reach the end of each matrix.
• Then, subtract each element of the second matrix from the corresponding element of the first matrix. For instance, if you have numbers a and b in corresponding positions, you calculate a - b.
• Do this for each element in the matrices.

In the given problem, after scalar multiplication of both matrices separately, the resultant matrices were subtracted element-wise to produce a new matrix. Keep in mind the order of subtraction as it affects the results significantly.

Element-Wise Operations

Element-wise operations refer to the process of performing operations on corresponding elements of matrices individually instead of as a whole. This approach is often necessary when performing operations like addition, subtraction, or multiplication as it applies to our given matrices.
Within the context of the exercise, we took the results from the scalar multiplication of two matrices and engaged in element-wise subtraction.

• Each position in the resulting matrix corresponds to an operation performed between the elements that occupy that same position in the two original matrices.
• This means element (1,1) from the first matrix interacts directly with element (1,1) from the second matrix, and so forth.

By utilizing element-wise operations, each corresponding element is influenced only by its pair in the other matrix. This is critical when matrices are used to represent real-world data like images or data grids where operations are performed pixel by pixel or data point by data point.