Problem 25 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$A=\left[\begin{array}{rr}2 & -5 \\\0 & 7\end{array}\right] \quad B=\left[\begin{array}{rrr}3 &\frac{1}{2} & 5 \\\1 & -1 & 3\end{array}\right] \quad C=\left[\begin{array}{rrr}2 & -\frac{5}{2} &0 \\\0 & 2 & -3\end{array}\right]$$ $$D=\left[\begin{array}{lll}7 & 3\end{array}\right] \quad E=\left[\begin{array}{l}1 \\\2 \\\0\end{array}\right] \quad F=\left[\begin{array}{lll}1 & 0 & 0 \\\0 & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$ $$G=\left[\begin{array}{rrr}5 & -3 & 10 \\\6 & 1 & 0 \\\\-5 & 2 & 2\end{array}\right] \quadH=\left[\begin{array}{rr}3 & 1 \\\2 & -1\end{array}\right]$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. (a) \(5 A\) (b) \(C-5 A\)

Step-by-Step Solution

Answer

(a) \(5A = \begin{bmatrix} 10 & -25 \\ 0 & 35 \end{bmatrix}\). (b) Cannot perform \(C - 5A\) due to dimension mismatch.

Step 1: Multiply Matrix by Scalar

First, we need to perform the operation for (a). Here, we have to multiply every element of the matrix \(A\) by the scalar 5. The matrix \(A\) is \[A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]. We multiply each entry:\[5A = 5 \cdot \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 5 \cdot 2 & 5 \cdot (-5) \ 5 \cdot 0 & 5 \cdot 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}.\]

Step 2: Check Dimensions for Subtraction

Next, for part (b), we must determine whether the matrices \(C\) and \(5A\) can be subtracted. Each matrix must have the same dimensions. Matrix \(C\) is of dimension 2x3: \[C = \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix}.\] Matrix \(5A\) is calculated as a 2x2 matrix: \[5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}.\] Since \(C\) is 2x3 and \(5A\) is 2x2, they have differing dimensions.

Step 3: Conclusion on Matrix Subtraction

Matrices \(C\) and \(5A\) cannot be subtracted because they do not share the same dimensions. Matrix subtraction requires that both matrices have the exact same number of rows and columns.

Key Concepts

Matrix Multiplication

Matrix multiplication can seem tricky at first, but it becomes simpler once you get the hang of it. Matrix multiplication is not like regular number multiplication. Instead, it involves the rows and columns of two matrices. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here is a step-by-step explanation.

Suppose you have two matrices, matrix A and matrix B, where A has dimensions of m x n, and B has dimensions of n x p. The resulting matrix, AB, will have dimensions m x p. This happens because each element in the resulting matrix is a sum of products between corresponding elements from the rows of A and the columns of B.

**Steps to Multiply Matrices:**

• Identify the dimensions of both matrices to ensure multiplication is possible.
• Multiply each element of the row of the first matrix by the corresponding element of the column in the second matrix.
• Sum these products to get the entry in the resulting matrix.
• Continue this process for each row and column until the entire matrix is filled.

This method is essential for many applications in mathematics, including solving systems of linear equations and transforming geometric figures.

Scalar Multiplication

Scalar multiplication is the process of multiplying a matrix by a single number called a scalar. This operation is much simpler than matrix multiplication because each element of the matrix is multiplied independently by the scalar without interaction between different elements.

Consider a matrix A and a scalar k. To perform scalar multiplication, simply multiply every element of matrix A by k. For example, if A is a 2x2 matrix such as:\[A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]and if your scalar is 5, the scalar multiplication looks like this:\[5A = 5 \cdot \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\]Every element of the matrix A is multiplied by 5.

Scalar multiplication is used in various contexts like scaling problems, where you need to increase or decrease values uniformly across a dataset, especially in matrices representing real-world objects, like vectors.

Matrix Subtraction

Matrix subtraction requires two matrices of the same size. That means they must have the same number of rows and columns. If matrix A is of dimension m x n, then matrix B must also be m x n to be subtracted from A. This condition is crucial because subtraction is performed element-wise.

Consider matrices C and 5A from our original exercise to understand why subtraction is not always possible. Matrix C has dimensions 2x3:\[C = \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix}\]And matrix 5A has dimensions 2x2:\[5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\]Since their dimensions do not match, we cannot perform subtraction. When matrices can be subtracted, each element in the resulting matrix is calculated by subtracting the corresponding element of the second matrix from the first.

Matrix subtraction is a straightforward process in terms of calculation, but the requirement for matching dimensions is a crucial rule to keep in mind.

Matrix Dimensions

Matrix dimensions tell us how many rows and columns a matrix has. Understanding dimensions is critical for performing matrix operations correctly. Dimensions are described in the form "m x n," where m is the number of rows, and n is the number of columns.

Here are some important things to note about matrix dimensions:

• If two matrices don't have the same dimensions, you cannot add or subtract them.
• Matrix multiplication requires the number of columns in the first matrix to be equal to the number of rows in the second matrix for the operation to be valid.

Typically, you would check dimensions before carrying out any operations to ensure everything is compatible. Let's take the original matrices C and 5A in the exercise for example. Matrix C is 2x3, meaning it has 2 rows and 3 columns. Matrix 5A is 2x2 with 2 rows and 2 columns.

Having an understanding of matrix dimensions will help avoid mistakes and facilitate more complex operations like finding an inverse or a determinant.