Problem 34 - College Algebra

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 operation is possible and \(A^2 = \begin{bmatrix} 200 & 300 \\ 75 & 725 \end{bmatrix}.\)

Step 1: Confirm Operation Possibility

To multiply two matrices together, the number of columns in the first matrix must match the number of rows in the second matrix. Since we are multiplying matrix \(A\) by itself (\(A^2 = A \cdot A\)), we need to confirm that \(A\) has the same number of columns as it has rows. Matrix \(A\) is a 2x2 matrix, hence the multiplication is possible.

Step 2: Matrix Multiplication Formula

For two 2x2 matrices \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(\begin{bmatrix} e & f \ g & h \end{bmatrix}\), the resulting matrix is computed as: \[ \begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{bmatrix} \] Apply this formula to calculate \(A^2\).

Step 3: Calculate Each Element of Resultant Matrix

Compute each element of \(A^2\).For element \((1,1)\), compute: \((-10)\times(-10) + 20\times5 = 100 + 100 = 200\).For element \((1,2)\), compute: \((-10)\times20 + 20\times25 = -200 + 500 = 300\).For element \((2,1)\), compute: \(5\times(-10) + 25\times5 = -50 + 125 = 75\).For element \((2,2)\), compute: \(5\times20 + 25\times25 = 100 + 625 = 725\).

Step 4: Write the Resultant Matrix

The resultant matrix \(A^2\) is:\[ \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix} \]

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a crucial component in the field of matrix algebra and refers to a square matrix with two rows and two columns. In a 2x2 matrix, we typically represent it in the following form:

• The first row contains two elements, referred to as the top left and top right values.
• The second row consists of two more elements, known as the bottom left and bottom right values.

For example, if we take the matrix\[\begin{bmatrix}-10 & 20 \ 5 & 25\end{bmatrix}\]this means:

• -10 is the first element in the first row and first column, which we call the "(1,1)" position.
• 20 is in the first row and second column, i.e., the "(1,2)" position.
• 5 is the element in the second row and first column, known as the "(2,1)" position.
• 25 is in the second row and second column, or the "(2,2)" position.

Understanding the layout and indices of a 2x2 matrix is essential for performing various mathematical operations, such as addition, subtraction, and importantly, multiplication.

Performing Matrix Operations

Matrix operations are the mathematical procedures we can perform on matrices, including addition, subtraction, and multiplication. The exercise primarily focuses on matrix multiplication. To multiply two matrices, the rule is simple yet crucial:

• The number of columns in the first matrix must equal the number of rows in the second matrix.

In our case, since we are looking at the matrix multiplication of matrix \(A\) with itself, i.e., \(A^2\):

• Matrix \(A\) is a 2x2 matrix with two rows and two columns.
• Since the rule is satisfied (columns of first matrix \(A\) are equal to the rows of second matrix \(A\)), multiplication is possible.

The formula for multiplying two 2x2 matrices \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] and \[ \begin{bmatrix} e & f \ g & h \end{bmatrix} \] results in:

• Top-left entry: \(ae + bg\)
• Top-right entry: \(af + bh\)
• Bottom-left entry: \(ce + dg\)
• Bottom-right entry: \(cf + dh\)

These calculations allow us to construct the resulting matrix from any 2x2 matrix multiplication and are integral to solving problems involving these structures.

The Nature of a Square Matrix

A square matrix, in simplest terms, is one that has an equal number of rows and columns. This property makes square matrices particularly interesting in the study of linear algebra.

• Common sizes for square matrices include 2x2, 3x3, and 4x4, but any size where the number of rows equals the number of columns qualifies as square.
• Square matrices hold special positions in matrix operations because they can be raised to powers (like \(A^2\), \(A^3\), etc.), which is useful in various applications such as transformations and solving systems of linear equations.
• Certain properties and operations, such as determinant and eigenvalues, are defined specifically for square matrices.

In the context of our exercise, matrix \(A\) is a square matrix since it has 2 rows and 2 columns, making it a 2x2 matrix. This characteristic makes it suitable for the operation \(A^2\), highlighting the versatility and importance of square matrices in matrix algebra.