Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A^{2}\)

Step-by-Step Solution

Answer

The matrix A squared is \( A^2 = \begin{bmatrix} 200 & 300 \\ 75 & 725 \end{bmatrix} \).

Step 1: Understand Matrix Multiplication

Before performing matrix multiplication, ensure that the matrices involved are compatible. Each element in the resulting matrix is computed as a dot product of the rows of the first matrix with the columns of the second matrix.

Step 2: Check Dimensions of A

Matrix multiplication is possible if the number of columns in the first matrix matches the number of rows in the second matrix. Matrix A is a 2x2 matrix.

Step 3: Set Up Multiplication for A^2

Since A is a 2x2 matrix, multiply A by itself: \[ A \text{ is } \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix}. \] The resulting matrix will also be 2x2.

Step 4: Calculate First Row of A^2

Compute the first row of the product matrix by the following calculations:- First Element: \((-10)(-10) + (20)(5) = 100 + 100 = 200\)- Second Element: \((-10)(20) + (20)(25) = -200 + 500 = 300\)

Step 5: Calculate Second Row of A^2

Compute the second row of the product matrix by the following calculations:- First Element: \((5)(-10) + (25)(5) = -50 + 125 = 75\)- Second Element: \((5)(20) + (25)(25) = 100 + 625 = 725\)

Step 6: Assemble Matrix A^2

Combine the computed elements into the resulting matrix:\[ A^2 = \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. The dimension of a matrix is defined by the number of rows and columns it contains. This is written as "rows x columns." For example, a 2x2 matrix has 2 rows and 2 columns. Knowing the dimensions allows us to determine whether certain operations, like matrix multiplication, are possible.

Matrix A, used in the exercise, has dimensions 2x2 which means it has 2 rows and 2 columns. Its form is:

• Row 1: [-10, 20]
• Row 2: [5, 25]

Matrix multiplication can only occur if the number of columns in the first matrix matches the number of rows in the second matrix. Therefore, two 2x2 matrices can multiply if their dimensions align properly. For example, matrix A can be multiplied by itself because both are 2x2.

Matrix Algebra

Matrix algebra considers operations that can be performed on matrices, such as addition, subtraction, and multiplication. Multiplication of matrices isn't as simple as multiplying individual numbers because it involves the dot product of rows and columns.

For multiplication to be defined in matrix algebra, the matrices must be compatible in dimensions as mentioned before. When multiplying two matrices, say matrix A and matrix B, you align one matrix's rows with another's columns. Calculate the product using the dot product method to derive a new matrix.

Matrix algebra not only deals with these multiplications but also with more complex operations like transposition and finding inverse matrices, which require specific mathematical procedures. In this exercise, we focus on multiplying a 2x2 matrix by itself, which provides a good foundation in understanding how matrix multiplication works in algebra.

Dot Product of Matrices

The dot product is a fundamental operation in matrix multiplication. To understand how we get the resulting elements in a product matrix, we need to understand the dot product between rows and columns of matrices.

Take two vectors (arrays of numbers): one from the row of the first matrix and the other from the column of the second matrix. Multiply the corresponding numbers and add the results together. This sum is the dot product, which becomes one element in the product matrix.

Let's look at matrix A from the exercise. To compute the first element of matrix \(A^2\), we perform the dot product of the first row of A with the first column of A:

• First element: \((-10) \times (-10) + 20 \times 5 = 100 + 100 = 200\)

This step-by-step dot product process is repeated for each element in the product matrix \(A^2\), resulting in a new matrix:

• Second element: \((-10) \times 20 + 20 \times 25 = -200 + 500 = 300\)
• Third element: \(5 \times (-10) + 25 \times 5 = -50 + 125 = 75\)
• Fourth element: \(5 \times 20 + 25 \times 25 = 100 + 625 = 725\)