Problem 46 - 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: \(\left.A^{2}=A \cdot A\right)\) \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{rrr}-2 & 3 & 4 \\ -1 & 1 & -5\end{array}\right], C=\left[\begin{array}{rr}0.5 & 0.1 \\ 1 & 0.2 \\ -0.5 & 0.3\end{array}\right], D=\left[\begin{array}{rrr}1 & 0 & -1 \\ -6 & 7 & 5 \\\ 4 & 2 & 1\end{array}\right]\) \(A^{2}\)

Step-by-Step Solution

Answer

The result of \( A^2 \) is \( \begin{bmatrix} 1 & 0 \\ 8 & 9 \end{bmatrix} \).

Step 1: Understand Matrix Multiplication Compatibility

The first step is to check if matrix multiplication is possible. Two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix. For calculating \( A^2 = A \cdot A \), we check if this condition is met for matrix \( A \). Matrix \( A \) is a 2x2 matrix. Therefore, it is possible to multiply \( A \) by itself.

Step 2: Write the Multiplication Expression

Write out the expression for multiplying \( A \) by itself: \[A \cdot A = \begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix}\]

Step 3: Calculate the First Element

Calculate the element in the first row and first column by multiplying the first row of the first matrix by the first column of the second matrix:\[1 \cdot 1 + 0 \cdot 2 = 1\]

Step 4: Calculate the Second Element

Calculate the element in the first row and second column by multiplying the first row of the first matrix by the second column of the second matrix:\[1 \cdot 0 + 0 \cdot 3 = 0\]

Step 5: Calculate the Third Element

Calculate the element in the second row and first column by multiplying the second row of the first matrix by the first column of the second matrix:\[2 \cdot 1 + 3 \cdot 2 = 8\]

Step 6: Calculate the Fourth Element

Calculate the element in the second row and second column by multiplying the second row of the first matrix by the second column of the second matrix:\[2 \cdot 0 + 3 \cdot 3 = 9\]

Step 7: Write the Resulting Matrix

Combine all the calculated elements to form the resulting matrix:\[A^2 = \begin{bmatrix} 1 & 0 \ 8 & 9 \end{bmatrix}\]

Key Concepts

Understanding Square Matrix

A square matrix is a type of matrix where the number of rows equals the number of columns. For example, a 2x2 matrix has two rows and two columns. Square matrices often play a vital role in various calculations and mathematical operations. They are especially significant when applying operations like squaring a matrix. In essence, when someone refers to squaring a matrix, they mean multiplying the matrix by itself.

• Square matrices can be of different sizes, such as 3x3, 4x4, etc.
• The identity matrix, a special type of square matrix, features ones on its diagonal and zeros elsewhere.
• Another example is the matrix used in this exercise, matrix \( A \), which is a 2x2 matrix.

Notice that square matrices like this one allow us to easily perform matrix multiplication, as they meet the required conditions for compatibility.

Matrix Compatibility Explained

Matrix compatibility is a crucial concept in matrix operations. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This rule ensures that each element in the resulting matrix is valid.

• If you want to multiply matrix \( A \) by matrix \( B \), ensure matrix \( A \)'s columns match matrix \( B \)'s rows.
• In the case of a square matrix like \( A \), compatibility is inherently satisfied for multiplication with itself.
• If matrices aren't compatible for multiplication, the operation will fail and produce no result.

Remember, compatibility is the first step to verify before attempting any matrix multiplication operations.

Basic Matrix Operations

Matrix operations encompass a variety of calculations performed on matrices. They include addition, subtraction, and multiplication, among others. Multiplication, which was the focus of this exercise, involves more intricate calculations than addition and subtraction. When multiplying matrices, the resulting matrix has dimensions defined by the row count of the first matrix and the column count of the second.

• Matrix\( A \) multiplied by itself in the exercise results in a new 2x2 matrix.
• Each element in the resulting matrix is derived from the dot product of corresponding rows and columns.
• Matrix multiplication isn't commutative, meaning \( A \times B eq B \times A \) in general.

Understanding these basic operations lays the groundwork for more complex applications in linear algebra and many real-world applications.

Exploring the 2x2 Matrix

In the world of matrices, a 2x2 matrix is the simplest form of a square matrix. It consists of two rows and two columns, making it a great starting point for those learning about matrix operations. The properties of a 2x2 matrix make calculations manageable while still offering valuable insight into more advanced mathematics.

• The determinant of a 2x2 matrix is calculated as \( ad-bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix.
• When squaring a matrix, as in this exercise, each element of the resulting matrix is calculated using the dot products of rows and columns in the original matrix.
• The simplicity of a 2x2 matrix allows for clear illustration of the principles behind more elaborate matrix operations.

By mastering the concepts around 2x2 matrices, students can build a foundational knowledge that is essential for delving into larger matrices and more complicated operations.