Problem 31 - 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) \(G F\) (b) \(G E\)

Step-by-Step Solution

Answer

(a) \(G F = G\), (b) \(GE = \begin{bmatrix}-1 \\\ 8 \\\ -1\end{bmatrix}\)

Step 1: Matrix Multiplication Compatibility for (a)

Determine if the matrices \(G\) and \(F\) can be multiplied. The matrix \(G\) is a \(3 \times 3\) matrix and \(F\) is also a \(3 \times 3\) matrix. For matrix multiplication \(A \times B\), the number of columns in \(A\) must equal the number of rows in \(B\). Since both are \(3 \times 3\), multiplication is possible.

Step 2: Perform Matrix Multiplication for (a)

Calculate the product \(G F\) by multiplying each element of the rows of \(G\) with the corresponding elements of the columns of \(F\) and summing the products:\[GF = \begin{bmatrix}5 & -3 & 10 \6 & 1 & 0 \-5 & 2 & 2 \\end{bmatrix} \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \\end{bmatrix} =\begin{bmatrix}5 & -3 & 10 \6 & 1 & 0 \-5 & 2 & 2 \\end{bmatrix}\]This is because multiplying by the identity matrix \(F\) results in the original matrix \(G\).

Step 3: Matrix Multiplication Compatibility for (b)

Determine if the matrices \(G\) and \(E\) can be multiplied. The matrix \(G\) is a \(3 \times 3\) matrix and \(E\) is a \(3 \times 1\) vector (or matrix). For multiplication to occur, the number of columns in \(G\) must equal the number of rows in \(E\), which is true (both have 3). Thus, multiplication is possible.

Step 4: Perform Matrix Multiplication for (b)

Calculate the product \(GE\) by multiplying each row of \(G\) with the column matrix \(E\):\[GE = \begin{bmatrix}5 & -3 & 10 \6 & 1 & 0 \-5 & 2 & 2 \\end{bmatrix} \begin{bmatrix}1 \2 \0\end{bmatrix} =\begin{bmatrix}5 \times 1 + (-3) \times 2 + 10 \times 0 \6 \times 1 + 1 \times 2 + 0 \times 0 \-5 \times 1 + 2 \times 2 + 2 \times 0\end{bmatrix} =\begin{bmatrix}-1 \8 \-1 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. A matrix's dimensions, denoted as \( m \times n \), tell us how many rows and columns it contains. For example, matrix \( A \) has dimensions \( 2 \times 2 \), meaning it has 2 rows and 2 columns.

It's important to remember the order of dimensions, as it affects compatibility in operations like addition and multiplication. Comparing dimensions helps determine if matrices can be combined. Often, simply knowing the dimensions is key to solving matrix-based problems.

Identity Matrix

An identity matrix is a square matrix that plays a special role in matrix operations, particularly multiplication. It is denoted usually by \( I \) and has 1s on the diagonal and 0s elsewhere.

For example, the identity matrix \( F \) in the exercise is a \( 3 \times 3 \) matrix:

• \( F = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \)

Multiplying any matrix by an identity matrix results in the original matrix. This property is similar to multiplying a number by 1. In the exercise, multiplying matrix \( G \) by \( F \) left \( G \) unchanged.

Matrix Compatibility

Matrix compatibility is a concept that determines if two matrices can be multiplied. This depends on their dimensions. For two matrices \( A \) and \( B \) to be compatible for multiplication, the number of columns in \( A \) must match the number of rows in \( B \).

Assessing compatibility is the first step before multiplication. In the exercise, both \( G \) and \( F \) are \( 3 \times 3 \) matrices, making them compatible. Similarly, matrix \( G \) and vector (or single-column matrix) \( E \), with dimensions \( 3 \times 3 \) and \( 3 \times 1 \) respectively, are also compatible.

Matrix Calculations

Matrix calculations involve performing various mathematical operations on matrices. Multiplication is one such operation and follows a systematic approach.

In multiplication, you multiply rows of the first matrix by columns of the second, summing the results to form each element of the product matrix. To illustrate, in the exercise, the product \( GE \) was calculated as follows:

• For each element in \( GE \), multiply corresponding elements from row of \( G \) and column of \( E \), then sum them up.
• Example from first row: \(5 \times 1 + (-3) \times 2 + 10 \times 0 = -1\).

Such systematic calculations ensure you get the right results every time.