Problem 15 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ A^{2}-B^{2} $$

Step-by-Step Solution

Answer

\(A^2 - B^2 = \begin{bmatrix} -8 & 16 \\ 25 & 3 \end{bmatrix}\).

Step 1: Understanding Matrix Squaring

Matrix squaring involves multiplying the matrix by itself. We must calculate \(A^2 = A \times A\) and \(B^2 = B \times B\).

Step 2: Calculating \(A^2\)

Given matrix \(A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\), calculate the product:\[A^2 = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 \times 1 + 2 \times 3 & 1 \times 2 + 2 \times 4 \ 3 \times 1 + 4 \times 3 & 3 \times 2 + 4 \times 4 \end{bmatrix} \]Calculating each element, we get:\[A^2 = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix}\]

Step 3: Calculating \(B^2\)

Given matrix \(B = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix}\), calculate the product:\[B^2 = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \times \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 0 \times 0 + (-3) \times (-5) & 0 \times (-3) + (-3) \times 2 \ -5 \times 0 + 2 \times (-5) & -5 \times (-3) + 2 \times 2 \end{bmatrix} \]Calculating each element, we get:\[B^2 = \begin{bmatrix} 15 & -6 \ -10 & 19 \end{bmatrix}\]

Step 4: Subtracting \(B^2\) from \(A^2\)

Now, subtract \(B^2\) from \(A^2\):\[A^2 - B^2 = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix} - \begin{bmatrix} 15 & -6 \ -10 & 19 \end{bmatrix} = \begin{bmatrix} 7-15 & 10-(-6) \ 15-(-10) & 22-19 \end{bmatrix}\]This results in:\[A^2 - B^2 = \begin{bmatrix} -8 & 16 \ 25 & 3 \end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra, where two matrices are combined to produce a third matrix. It is essential to understand that matrix multiplication is not the same as multiplying single numbers. For multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.

The process involves taking rows from the first matrix and columns from the second matrix, multiplying corresponding elements, and then summing the products to get the resultant matrix's elements. Here's a breakdown of how it works:

• Select a row from the first matrix.
• Select a column from the second matrix.
• Multiply each pair of elements from the row and column positions.
• Add all the resulting products together to get the single element of the new matrix.

There are important rules in matrix multiplication, such as the commutative property does not apply, meaning the order in which you multiply matrices matters. For example, multiplying matrix A by matrix B is not necessarily the same as multiplying matrix B by matrix A. Understanding these principles is crucial when working with matrix operations in advanced mathematics.

Matrix Subtraction

Matrix subtraction is a straightforward operation compared to multiplication and follows similar rules to basic arithmetic subtraction but on a matrix level. To subtract one matrix from another, both matrices must be of the same dimension, meaning they must have the same number of rows and columns.

The process is simple:

• Ensure that both matrices are of the same size.
• Subtract each element of the second matrix from the corresponding element of the first matrix. This is done element-wise, meaning element by element at each corresponding position.

For example, if you have matrices A and B:\[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]The subtraction A - B is calculated as:\[A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix}\]This method ensures that each pair of corresponding elements are subtracted to form the resulting matrix. Being clear on matrix dimensions and aligning elements correctly is key to correctly performing matrix subtraction.

Matrix Algebra

Matrix algebra encompasses operations like addition, subtraction, and multiplication of matrices, and is a cornerstone of linear algebra. It is especially useful in areas such as system of equations, computer graphics, and economics.

Some key concepts in matrix algebra include:

• Commutative Property: As mentioned, it does not hold for multiplication but does for addition, meaning A + B = B + A.
• Associative Property: This holds for both addition and multiplication, allowing for grouping operations without worrying about the order, ((A + B) + C = A + (B + C)) and ((A × B) × C = A × (B × C)).
• Identity Matrix: In multiplication, there exists an identity matrix such that any matrix multiplied by the identity matrix returns the original matrix (I × A = A).
• Inverse Matrix: Some matrices may have an inverse (denoted as A^-1), which, when multiplied by the original matrix, results in the identity matrix (A × A^-1 = I).

Understanding these properties is essential as they provide the foundational rules that allow matrices to perform complex operations and transformations in various mathematical contexts. Mastery of matrix algebra opens doors to solving real-world problems that are represented in matrix form.