Problem 27 - 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) \(A H\) (b) \(H A\)

Step-by-Step Solution

Answer

(a) \(AH = \begin{bmatrix} -4 & 7 \\ 14 & -7 \end{bmatrix}\); (b) \(HA = \begin{bmatrix} 6 & -8 \\ 4 & -17 \end{bmatrix}\)."

Step 1: Determine if Matrix Multiplication is Possible for AH

Matrix multiplication is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix. For \(A\) and \(H\), \(A\) is a \(2 \times 2\) matrix and \(H\) is also a \(2 \times 2\) matrix. Therefore, multiplication is possible.

Step 2: Multiply Matrices A and H

Multiply matrices \(A\) and \(H\) by taking the dot product of rows of \(A\) with columns of \(H\). Calculate the elements of the resulting matrix:- First row, first column: \((2 \cdot 3) + (-5 \cdot 2) = 6 - 10 = -4\)- First row, second column: \((2 \cdot 1) + (-5 \cdot -1) = 2 + 5 = 7\)- Second row, first column: \((0 \cdot 3) + (7 \cdot 2) = 0 + 14 = 14\)- Second row, second column: \((0 \cdot 1) + (7 \cdot -1) = 0 - 7 = -7\)The resulting matrix is \(\begin{bmatrix} -4 & 7 \ 14 & -7 \end{bmatrix}\).

Step 3: Determine if Matrix Multiplication is Possible for HA

For \(H A\), \(H\) is a \(2 \times 2\) matrix and \(A\) is also a \(2 \times 2\) matrix. The number of columns in \(H\) equals the number of rows in \(A\), making the multiplication possible.

Step 4: Multiply Matrices H and A

Multiply matrices \(H\) and \(A\) by taking the dot product of rows of \(H\) with columns of \(A\).Calculate the elements of the resulting matrix:- First row, first column: \((3 \cdot 2) + (1 \cdot 0) = 6 + 0 = 6\)- First row, second column: \((3 \cdot -5) + (1 \cdot 7) = -15 + 7 = -8\)- Second row, first column: \((2 \cdot 2) + (-1 \cdot 0) = 4 + 0 = 4\)- Second row, second column: \((2 \cdot -5) + (-1 \cdot 7) = -10 - 7 = -17\)The resulting matrix is \(\begin{bmatrix} 6 & -8 \ 4 & -17 \end{bmatrix}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation that is often used in mathematics, physics, computer science, and engineering. This operation involves combining two matrices to produce a new matrix. It is important to remember that not every pair of matrices can be multiplied together. In order for matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.
For example, suppose we have matrices \( A \) and \( H \) both as \( 2 \times 2 \) matrices. Since the number of columns in \( A \) (which is 2) matches the number of rows in \( H \) (also 2), this operation is possible.
When multiplying matrices \( A \) and \( H \), we perform a series of dot products between the rows of \( A \) and the columns of \( H \). Each dot product result corresponds to an element in the new matrix.

• Multiply the first row of \( A \) with the first column of \( H \) to get the first element of the new first row.
• Continue this for all row-column combinations to fill the resulting matrix.

The product of matrix multiplication is another matrix, and in our example, the resulting matrix has the same dimensions as the original square matrices involved in the multiplication process.

Dot Product

The dot product is a fundamental component used in matrix multiplication. It involves multiplying corresponding elements from two lists of numbers (a row and a column) and then summing these products. This process is repeated for each row-column pairing in matrix multiplication.
Take for instance the following dot product calculation from the step-by-step solution: when we multiply matrices \( A \) and \( H \), the element results from multiplying the elements of a row from \( A \) with elements of a column from \( H \) and summing these products. For example:

• For the element in the first row, first column of the resulting matrix from \( A \) and \( H \): \((2 \cdot 3) + (-5 \cdot 2) = 6 - 10 = -4\).

The process involves simple arithmetic, yet it is crucial for ensuring each number in the resulting matrix is accurately calculated. Proper application of the dot product is critical for correct matrix multiplication results.

Matrix Dimensions

Matrix dimensions are pivotal when it comes to understanding various matrix operations, including multiplication. The dimensions of a matrix are defined by the number of its rows and columns, typically expressed as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
When working with matrices, always pay close attention to these dimensions as they dictate the eligibility of certain operations. A matrix with dimensions 3x2 cannot be multiplied with a 2x4 matrix because the inner dimensions (2 in the first and 2 in the second ) align, allowing multiplication to be performed.
For example, in the matrices \( A \) and \( H \) used earlier, both with dimensions \( 2 \times 2 \), this criterion for multiplication is satisfied because the number of columns in \( A \) equals the number of rows in \( H \). It is always beneficial to ensure matrix dimensions are compatible before proceeding with operations, avoiding computational errors and making calculations seamless.