Problem 45 - 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: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ $$ D^{2} $$

Step-by-Step Solution

Answer

Matrix \( D^2 \) is \( \begin{bmatrix} -3 & -2 & -2 \\ 63 & 59 & 46 \\ -4 & 16 & 7 \end{bmatrix} \).

Step 1: Understand Matrix Squaring

To square a matrix, we need to multiply that matrix by itself. The given matrix is \( D \), which we need to square to find \( D^2 \). Matrix \( D \) is a 3x3 matrix, so we will perform matrix multiplication of \( D \) by \( D \).

Step 2: Set Up the Multiplication

Matrix \( D \) is given by \[D = \begin{bmatrix}1 & 0 & -1 \-6 & 7 & 5 \4 & 2 & 1\end{bmatrix}\]To find \( D^2 \), we need to set up the product \( D \cdot D \).

Step 3: Compute Elements of the Resultant Matrix

To find each element \( d_{ij} \) of the resultant matrix \( D^2 \), take the dot product of the \( i \)-th row of matrix \( D \) and the \( j \)-th column of matrix \( D \). For example, to find the element in the first row, first column of \( D^2 \): \[ d_{11} = (1 \times 1) + (0 \times -6) + (-1 \times 4) = 1 - 4 = -3 \]

Step 4: Calculate Each Entry

Continue the process for all elements:- \( d_{12} = (1 \times 0) + (0 \times 7) + (-1 \times 2) = 0 - 2 = -2 \)- \( d_{13} = (1 \times -1) + (0 \times 5) + (-1 \times 1) = -1 - 1 = -2 \)- \( d_{21} = (-6 \times 1) + (7 \times -6) + (5 \times 4) = -6 + 49 + 20 = 63 \)- \( d_{22} = (-6 \times 0) + (7 \times 7) + (5 \times 2) = 49 + 10 = 59 \)- \( d_{23} = (-6 \times -1) + (7 \times 5) + (5 \times 1) = 6 + 35 + 5 = 46 \)- \( d_{31} = (4 \times 1) + (2 \times -6) + (1 \times 4) = 4 - 12 + 4 = -4 \)- \( d_{32} = (4 \times 0) + (2 \times 7) + (1 \times 2) = 14 + 2 = 16 \)- \( d_{33} = (4 \times -1) + (2 \times 5) + (1 \times 1) = -4 + 10 + 1 = 7 \)

Step 5: Write the Resulting Matrix

After computing all elements, the matrix \( D^2 \) is:\[D^2 = \begin{bmatrix}-3 & -2 & -2 \63 & 59 & 46 \-4 & 16 & 7\end{bmatrix}\]This is the product of \( D \cdot D \), and hence \( D^2 \).

Key Concepts

Matrix Squaring

Matrix squaring is all about multiplying a matrix with itself. When we take a matrix, say \( D \), and multiply it by itself, the process is called squaring the matrix to find \( D^2 \). This involves matrix multiplication, where each element of the resulting matrix is computed by taking the dot product of rows and columns from the original matrix. It's just like multiplying numbers, but you do it for rows and columns!
Squaring a matrix is possible only for square matrices (like our 3x3 example). These are matrices with an equal number of rows and columns, which allows the multiplication to happen smoothly.

Dot Product

Understanding the dot product is key to matrix multiplication and squaring. The dot product involves multiplying corresponding elements of a row from the first matrix with a column from the same matrix, and then summing those products. For example, if matrix \( D \) has elements in its first row as \([1, 0, -1]\) and its first column as \([1, -6, 4]\), then:

• The dot product for this row and column will be \((1 \times 1) + (0 \times -6) + (-1 \times 4) = 1 + 0 - 4 = -3\).

This dot product would become the first element of the resulting matrix. Repeat this process for all rows and columns to fill out the entire squared matrix.

Matrix Operations

Matrix operations such as addition, subtraction, and multiplication require careful alignment of dimensions. For matrix multiplication specifically, the number of columns in the first matrix should match the number of rows in the second one. In matrix squaring, since you're multiplying a matrix by itself, this condition is automatically satisfied.
While performing these operations, you need to align elements precisely and follow the order of operations meticulously. Always work row by column, sticking to each corresponding element operation to ensure accuracy.

• Remember: matrix multiplication is generally not commutative (\( A \times B eq B \times A \)). This rule is crucial so don't swap the order of multiplication in operations.

3x3 Matrix

A 3x3 matrix is a matrix with three rows and three columns. It's a kind of square matrix, which is advantageous for operations like squaring since the number of rows equals the number of columns.
Each position in a 3x3 matrix is outlined as an element that can be identified by two indexes: row number and column number. When you manipulate a 3x3 matrix, you need to handle nine elements in total.

• For squaring a 3x3 matrix, you'll perform multiplication and summation for each element, ensuring compatibility across rows and columns to compute each part of the resultant matrix correctly.
• 3x3 matrices sit at a sweet spot between complexity and manageability, often appearing in operations needing more detail than 2x2 matrices but not as heavy as larger matrices.