Problem 25 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$\left[\begin{array}{rrrr} 9 & 4 & 1 & -2 \\ 5 & -6 & 3 & 4 \\ 2 & -5 & 1 & 2 \end{array}\right]-\left[\begin{array}{rrrr} -2 & 5 & 1 & 3 \\ 0 & 1 & 0 & 2 \\ -8 & 3 & 2 & 1 \end{array}\right]+\left[\begin{array}{rrrr} 2 & 4 & 0 & 3 \\ 4 & -5 & 1 & 6 \\ 2 & -3 & 0 & 8 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 13 & 3 & 0 & -2 \\ 9 & -12 & 4 & 8 \\ 12 & -11 & -1 & 9 \end{bmatrix} \).

Step 1: Matrix Dimensions Verification

Verify if all matrices have the same dimensions. Each given matrix is a 3x4 matrix. Since they all have the same dimensions, we can proceed with the addition and subtraction.

Step 2: Perform Matrix Subtraction

Subtract the second matrix from the first matrix. So, perform the element-wise subtraction: \[\begin{bmatrix}9 & 4 & 1 & -2 \5 & -6 & 3 & 4 \2 & -5 & 1 & 2\end{bmatrix} - \begin{bmatrix}-2 & 5 & 1 & 3 \0 & 1 & 0 & 2 \-8 & 3 & 2 & 1\end{bmatrix} = \begin{bmatrix}9 - (-2) & 4 - 5 & 1 - 1 & -2 - 3 \5 - 0 & -6 - 1 & 3 - 0 & 4 - 2 \2 - (-8) & -5 - 3 & 1 - 2 & 2 - 1\end{bmatrix} = \begin{bmatrix}11 & -1 & 0 & -5 \5 & -7 & 3 & 2 \10 & -8 & -1 & 1\end{bmatrix}\]

Step 3: Perform Matrix Addition

Add the resulting matrix from Step 2 to the third matrix. Perform the element-wise addition:\[\begin{bmatrix}11 & -1 & 0 & -5 \5 & -7 & 3 & 2 \10 & -8 & -1 & 1\end{bmatrix} + \begin{bmatrix}2 & 4 & 0 & 3 \4 & -5 & 1 & 6 \2 & -3 & 0 & 8\end{bmatrix} = \begin{bmatrix}11 + 2 & -1 + 4 & 0 + 0 & -5 + 3 \5 + 4 & -7 - 5 & 3 + 1 & 2 + 6 \10 + 2 & -8 - 3 & -1 + 0 & 1 + 8\end{bmatrix} = \begin{bmatrix}13 & 3 & 0 & -2 \9 & -12 & 4 & 8 \12 & -11 & -1 & 9\end{bmatrix}\]

Step 4: Final Verification

Verify the dimensions and operation consistency of the resulting matrix. The result is a 3x4 matrix, consistent with the dimensions of the input matrices.

Key Concepts

Matrix Subtraction

Matrix subtraction is an operation where you subtract corresponding elements of two matrices to produce a new matrix. To perform matrix subtraction, the matrices involved must have the same dimensions. This ensures that every element in one matrix has a corresponding element in the other matrix to subtract.

Here's how matrix subtraction works:

• Identify corresponding elements from each matrix.
• Subtract each element in the second matrix from the corresponding element in the first matrix.
• Write the result in the corresponding position in the new matrix.

For instance, if you have two matrices, \[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]The subtraction of matrix B from A (i.e., \(A - B\)) is calculated element-wise as:\[A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix}\]Matrix subtraction, like addition, is straightforward as long as the dimensions are compatible. It's as simple as subtracting numbers, but done in a way that respects the structure of the matrices.

Matrix Addition

Matrix addition involves adding two matrices by adding their corresponding elements. Like subtraction, addition requires that the matrices have the same dimensions. If dimensions mismatch, the operation cannot be performed.

When performing matrix addition:

• Ensure the matrices share the same order of dimensions, say both are 2x2 or both are 3x4.
• Add the corresponding elements together to form a new matrix.

For example, with two matrices:\[C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix}, D = \begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} \end{bmatrix}\]Their addition \(C + D\) is given by:\[C + D = \begin{bmatrix} c_{11} + d_{11} & c_{12} + d_{12} \ c_{21} + d_{21} & c_{22} + d_{22} \end{bmatrix}\]The sum of matrices is calculated by adding element-wise. This operation forms a new matrix composed of the sum of each pair of corresponding elements from the two matrices.

Matrix Dimensions Verification

Matrix dimensions verification is a crucial step before performing any operations involving multiple matrices such as addition or subtraction. It assures that the operations are feasible and that the matrices involved are "conformable."

To verify matrix dimensions:

• Check each matrix to determine its dimensions. A matrix's dimensions are written as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
• Ensure each matrix involved in the operation has identical dimensions. For example, both should be 2x3 or both should be 4x4, as in our exercise where all matrices were 3x4.

Performing a matrix operation without verifying dimensions can lead to errors. For example, adding a 2x3 matrix to a 3x3 matrix is not possible since the dimensions are not the same. This step guarantees the mathematical validity of further matrix operations and prevents unforeseeable calculation errors.