Problem 9 - College Algebra

Question

Use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{rr}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{rr}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{rr}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{rr}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{rr}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) \(B-E\)

Step-by-Step Solution

Answer

The result of \(B - E\) is \(\begin{bmatrix} -4 & 2 \\ 8 & 1 \end{bmatrix}\).

Step 1: Determine Matrix Dimensions

First, let's determine the dimensions of matrices \(B\) and \(E\). Matrix \(B\) is \(2 \times 2\), which means it has 2 rows and 2 columns. Similarly, matrix \(E\) is also \(2 \times 2\). Since both matrices are of the same size, we can proceed with the subtraction operation.

Step 2: Subtract the Matrices

To subtract matrices \(B\) and \(E\), subtract each corresponding element in \(E\) from the element in \(B\). \[B - E = \begin{bmatrix} 2 & 14 \ 22 & 6 \end{bmatrix} - \begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix}\]Performing the subtraction element by element, we get:\[B - E = \begin{bmatrix} 2-6 & 14-12 \ 22-14 & 6-5 \end{bmatrix} = \begin{bmatrix} -4 & 2 \ 8 & 1 \end{bmatrix}\]

Step 3: Verify the Result

Let's verify the result by checking our calculations. For the element in the first row, first column: \(2 - 6 = -4\). For the first row, second column: \(14 - 12 = 2\). For the second row, first column: \(22 - 14 = 8\). Lastly, for the second row, second column: \(6 - 5 = 1\). Each calculation matches our derived matrix.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra where two matrices with the same dimensions are added together. The process is quite intuitive. You simply add each corresponding element of the matrices. For example, given two matrices, say matrix G and matrix H:

• Matrix G: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
• Matrix H: \[\begin{bmatrix} e & f \ g & h \end{bmatrix}\]

The addition of G and H can be expressed as:\[G + H = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\]This operation is only defined when both G and H have the same number of rows and columns. It's essential to check the matrix dimensions before attempting addition.

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead, you subtract elements from the first matrix by the corresponding elements in the second matrix. Like addition, subtraction requires matrices to have the same dimensions. Let's look at the subtraction of matrices B and E:

• Matrix B: \[\begin{bmatrix} 2 & 14 \ 22 & 6 \end{bmatrix}\]
• Matrix E: \[\begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix}\]

Start by subtracting each element in E from the corresponding element in B:

• First row, first column: 2 - 6 = -4
• First row, second column: 14 - 12 = 2
• Second row, first column: 22 - 14 = 8
• Second row, second column: 6 - 5 = 1

So, the resulting matrix is:\[\begin{bmatrix} -4 & 2 \ 8 & 1 \end{bmatrix}\]As with addition, always verify that the matrices you're subtracting are the same size; otherwise, the operation is undefined.

Matrix Dimensions

Understanding matrix dimensions is crucial when performing matrix operations like addition and subtraction. The dimensions of a matrix are given in terms of rows and columns. For instance, a matrix with 3 rows and 2 columns is a 3x2 matrix. This information tells you how many elements are in each matrix and, importantly, whether certain operations are possible.For a clear picture:

• Matrix A: \[\begin{bmatrix} a & b \ c & d \ e & f \end{bmatrix} \] is 3x2.
• Matrix B: \[\begin{bmatrix} x & y \ z & w \end{bmatrix}\] is 2x2.

Operations like addition or subtraction require that matrices have identical dimensions. Therefore, you couldn't add or subtract A from B in the example above, as their sizes do not match. Recognizing this can save a lot of time and confusion.

Matrix Algebra

Matrix algebra encompasses a range of operations beyond simple addition and subtraction, including multiplication and finding determinants. When dealing with matrices, understanding their properties is crucial because they follow different rules than regular numbers. Some key points in matrix algebra include:

• Matrix Multiplication: Unlike addition and subtraction, multiplication doesn't necessarily require same-sized matrices. Instead, the number of columns in the first matrix must equal the number of rows in the second.
• Determinants and Inverses: These are properties specific to square matrices (where rows equal columns). They play a significant role in solving systems of linear equations.

Matrix algebra is a powerful tool used across physics, computer science, and economics, providing a systematic method to manipulate linear equations. Mastery of matrix operations opens doors to understanding more complex mathematical concepts like eigenvalues and transformation matrices.