Problem 114 - Precalculus

Question

Apply a graphing utility to perform the indicated matrix operations, if possible. $$A=\left[\begin{array}{rrrr}1 & 7 & 9 & 2 \\\\-3 & -6 & 15 & 11 \\\0 & 3 & 2 & 5 \\\9 & 8 & -4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}7 & 9 \\\8 & 6 \\\\-4 & -2 \\\3 & 1\end{array}\right]$$ $$A A$$

Step-by-Step Solution

Answer

The product \( AA \) is \( \begin{bmatrix} -2 & -26 & 113 & 52 \\ 0 & 9 & 195 & 67 \\ 15 & 33 & -34 & 34 \\ 70 & 99 & 117 & 150 \end{bmatrix} \).

Step 1: Understand Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each element in the resulting matrix is obtained by taking the dot product of a row in the first matrix and a column in the second matrix.

Step 2: Check Matrix Dimensions

Matrix \( A \) is a \( 4 \times 4 \) matrix. Since we are multiplying \( A \) by itself, \( A \times A \) is possible because both matrices have the same dimension, \( 4 \times 4 \).

Step 3: Set Up Matrix Multiplication

To compute \( A \times A \), we need to determine each element \( c_{ij} \) in the resulting matrix \( C \), where each element is the sum of products of elements from the \( i \)-th row of the first \( A \) and \( j \)-th column of the second \( A \).

Step 4: Calculate Each Element of Resulting Matrix

For example, the element at position \( c_{11} \) is found by multiplying the elements of the first row of the first \( A \) with the first column of the second \( A \) and summing them: \( c_{11} = 1 imes 1 + 7 imes (-3) + 9 imes 0 + 2 imes 9 = 1 - 21 + 0 + 18 = -2 \). Repeat this operation for all elements.

Step 5: Complete Matrix Multiplication

Continue calculating all elements similarly. The result after calculations is:\[A A = \begin{bmatrix} -2 & -26 & 113 & 52 \ 0 & 9 & 195 & 67 \ 15 & 33 & -34 & 34 \ 70 & 99 & 117 & 150 \end{bmatrix}\]

Step 6: Interpret the Result

The matrix \( C \) is a \( 4 \times 4 \) matrix, which represents the product \( A \times A \). This operation confirms that the square of the matrix is computed accurately.

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial when it comes to matrix multiplication. Each matrix is represented as 'rows by columns'. For example, matrix \( A \) is a \( 4 \times 4 \) matrix, meaning it has 4 rows and 4 columns.
A key rule of matrix multiplication is that the number of columns in the first matrix must match the number of rows in the second matrix.

• For instance, to multiply matrix \( A \) by itself, the number of columns in the first \( A \) (which is 4) must equal the number of rows in the second \( A \) (also 4).
• Therefore, \( A \times A \) is possible since both matrices have compatible dimensions.
  

Understanding dimensions is the first step in ensuring that a matrix multiplication operation is valid. It's like a compatibility check before diving into calculations.

Dot Product

The concept of the dot product is essential when determining each element of the resulting matrix from two multiplied matrices. When multiplying matrices, each entry of the resulting matrix is calculated using the dot product of corresponding rows and columns.
Let's break down how this works:

• For any element \( c_{ij} \) of the resulting matrix \( C \), it is computed by multiplying corresponding elements from the \( i \)-th row of the first matrix and the \( j \)-th column of the second matrix, then summing those products.
  
• For example, if you need to find the element in the first row and first column of the result, you take each number from the first row of the first \( A \) and multiply it by the matching number from the first column of the second \( A \), then add those products:

\[c_{11} = (1 \cdot 1) + (7 \cdot -3) + (9 \cdot 0) + (2 \cdot 9) = -2\]This arithmetic forms the building blocks of matrix multiplication.

Resulting Matrix

The resulting matrix is the outcome of the matrix multiplication process. It embodies all the calculated dot products put together.
In our case, the product \( A \times A \) results in a \( 4 \times 4 \) matrix. This is because both \( A \) matrices involved in the multiplication are of size \( 4 \times 4 \).

• Every element of this resulting matrix is a dot product result, as calculated in each step.
  
• The final matrix therefore is:\[A \times A = \begin{bmatrix} -2 & -26 & 113 & 52 \ 0 & 9 & 195 & 67 \ 15 & 33 & -34 & 34 \ 70 & 99 & 117 & 150 \end{bmatrix}\]

This matrix represents all the intersections of elements processed through dot products, showcasing the complexity yet pattern-oriented nature of linear algebra operations.