Problem 29 - Precalculus Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$\begin{aligned} &A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right]\\\ &D=\left[\begin{array}{llll} 7 & 3 \end{array}\right] \quad E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \quad F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\\ &G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \quad H=\left[\begin{array}{rr} 3 & 1 \\ 2 & -1 \end{array}\right] \end{aligned}$$ 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) Result is the same as \(G\); (b) Result is \([-1, 8, -1]^T\).

Step 1: Check Multiplicability of Matrices for (a)

The operation asks for the multiplication of matrix \(G\) and matrix \(F\). For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(G\) is a 3x3 matrix, and matrix \(F\) is also a 3x3 matrix. Since the number of columns in \(G\) (which is 3) matches the number of rows in \(F\) (also 3), we can perform this multiplication.

Step 2: Carry Out Matrix Multiplication for (a)

Since both \(G\) and \(F\) are 3x3 matrices, their multiplication will result in another 3x3 matrix. Matrix \(F\) is the identity matrix; thus, multiplying any matrix by the identity matrix results in the original matrix. Therefore, the result of \(G F\) is simply matrix \(G\):\[\left[ \begin{array}{rrr}5 & -3 & 10 \6 & 1 & 0 \-5 & 2 & 2\end{array} \right]\]

Step 3: Check Multiplicability of Matrices for (b)

For the operation \(G E\), we again check the dimensions. Matrix \(G\) is 3x3, and matrix \(E\) is 3x1. The number of columns in \(G\) (3) matches the number of rows in \(E\) (3), so the multiplication can be performed.

Step 4: Perform Matrix Multiplication for (b)

Multiply matrix \(G\) with matrix \(E\). This will result in a new 3x1 matrix. Each element in the resulting matrix is the dot product of the corresponding row of \(G\) and column of \(E\):\[\left[ \begin{array}{r}5 \cdot 1 + (-3) \cdot 2 + 10 \cdot 0 \6 \cdot 1 + 1 \cdot 2 + 0 \cdot 0 \-5 \cdot 1 + 2 \cdot 2 + 2 \cdot 0\end{array} \right] = \left[ \begin{array}{r}5 - 6 + 0 \6 + 2 + 0 \-5 + 4 + 0\end{array} \right] =\left[ \begin{array}{r}-1 \8 \-1\end{array} \right]\]

Key Concepts

Matrix Dimension Compatibility

When it comes to matrix multiplication, an essential concept to understand is matrix dimension compatibility. This is a rule that determines when two matrices can be multiplied together. For a matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix.

Let's break this down further using small, concrete steps:

• If you have matrix \(A\) with dimensions \(m \times n\), it has \(m\) rows and \(n\) columns.
• Matrix \(B\) must have \(n\) rows, meaning that it should fit with \(n\), the number of columns in \(A\), for multiplication.
• If \(B\) has \(k\) columns, the resulting matrix will have dimensions \(m \times k\).

For example, in our exercise for part (a), both matrix \(G\) and matrix \(F\) are of size \(3 \times 3\). Since the number of columns of \(G\) (which is 3) equals the number of rows of \(F\) (also 3), they can indeed be multiplied, following our compatibility check. Similarly, for part (b), matrix \(G\) has 3 columns, and matrix \(E\) has 3 rows, deeming them compatible too.

Matrix Identity Property

The identity property of matrices is a fundamental concept in linear algebra. An identity matrix, often denoted as \(I\), has specific properties that make it akin to the number 1 in scalar multiplication. Let's explore how this works:

• An identity matrix has 1s on its diagonal (from top-left to bottom-right) and 0s elsewhere.
• When any matrix \(A\) is multiplied by the identity matrix \(I\) of compatible dimensions, the product is the original matrix \(A\). This is because multiplying by 1 (elements on the diagonal) keeps values unchanged, while multiplying by 0 nullifies others.

In the exercise example, matrix \(F\) is the identity matrix of size \(3 \times 3\). When matrix \(G\) is multiplied by \(F\), the result remains matrix \(G\), illustrating the identity matrix's effect. Use the identity matrix when you wish to retain the original matrix in operations, much like multiplying a number by 1.

Dot Product in Matrices

The dot product plays a crucial role when performing matrix multiplication. Understanding this concept is key to calculating each element of the resulting matrix:

Here's what happens in a dot product during matrix multiplication:

• You take a row from the first matrix and a column from the second matrix.
• For each corresponding pair of elements from these row and column, you multiply them together.
• Sum all these products together to get a single number—this is the dot product, and it becomes one element of the resulting matrix.

Considering part (b) of the exercise, when multiplying matrix \(G\) with matrix \(E\), each element of the new matrix is derived from the dot product of each row in \(G\) with the column in \(E\). For instance, the first element of the result is calculated as: \(5 \cdot 1 + (-3) \cdot 2 + 10 \cdot 0 = -1\). This process repeats for each row in \(G\), providing a concise but powerful method for matrix manipulation.