Problem 23 - College Algebra with Modeling and Visualization

Question

Evaluate the matrix expression. $$\left[\begin{array}{rr}4 & 6 \\\3 & -7\end{array}\right]-2\left[\begin{array}{rr}1 & 0 \\\\-4 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The evaluated matrix is \( \begin{bmatrix} 2 & 6 \\ 11 & -9 \end{bmatrix} \).

Step 1: Identify the Matrices and Scalar Multiplier

The first matrix is \( \begin{bmatrix} 4 & 6 \ 3 & -7 \end{bmatrix} \) and the second matrix is \( \begin{bmatrix} 1 & 0 \ -4 & 1 \end{bmatrix} \). The scalar multiplier for the second matrix is 2.

Step 2: Multiply the Second Matrix by the Scalar

Multiply each element in the matrix \( \begin{bmatrix} 1 & 0 \ -4 & 1 \end{bmatrix} \) by the scalar 2. This gives:\[ 2 \times \begin{bmatrix} 1 & 0 \ -4 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times 1 & 2 \times 0 \ 2 \times (-4) & 2 \times 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \ -8 & 2 \end{bmatrix} \]

Step 3: Subtract the Resulting Matrix from the First Matrix

Subtract each corresponding element of the matrix \( \begin{bmatrix} 2 & 0 \ -8 & 2 \end{bmatrix} \) from \( \begin{bmatrix} 4 & 6 \ 3 & -7 \end{bmatrix} \):\[ \begin{bmatrix} 4 & 6 \ 3 & -7 \end{bmatrix} - \begin{bmatrix} 2 & 0 \ -8 & 2 \end{bmatrix} = \begin{bmatrix} 4 - 2 & 6 - 0 \ 3 - (-8) & -7 - 2 \end{bmatrix} \]Simplifying, we find:\[ \begin{bmatrix} 2 & 6 \ 11 & -9 \end{bmatrix} \]

Step 4: Verify the Final Result

Revisit each operation to ensure all mathematical steps have been carried out correctly. For matrix subtraction, ensure that all components match, resulting in the final expression:\[ \begin{bmatrix} 2 & 6 \ 11 & -9 \end{bmatrix} \]

Key Concepts

Matrix Subtraction

Matrix subtraction is the process of deducting corresponding elements of one matrix from another matrix of the same dimensions. Both matrices need to have the same number of rows and columns, otherwise, subtraction cannot be performed.
\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} - \begin{bmatrix} e & f \ g & h \end{bmatrix} = \begin{bmatrix} a-e & b-f \ c-g & d-h \end{bmatrix} \]
This operation is similar to simple arithmetic subtraction, but it applies to each individual element in the matrices.

• Ensure that each matrix is of the same size before subtracting.
• Line up each element and perform the arithmetic operation.
• Subtraction results in a new matrix that is the same size as the original two.

Matrix subtraction is straightforward but requires attention to detail in ensuring proper subtraction of corresponding elements.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant or scalar. This operation scales the entire matrix, making all elements proportionally larger or smaller depending on the scalar.
For a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and a scalar \( s \), the multiplication is expressed as:
\[ s \times \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} sa & sb \ sc & sd \end{bmatrix} \]

• Multiply each element of the matrix by the scalar independently.
• The matrix retains its original dimensions after scalar multiplication.
• Multiplying by 1 leaves the matrix unchanged, while multiplying by 0 gives a zero matrix.

Scalar multiplication is a fundamental operation as it alters matrix elements consistently without changing their positions. It's crucial when matrices are intended to be scaled for further calculations.

Matrix Arithmetic

Matrix arithmetic encompasses fundamental operations like addition, subtraction, and multiplication of matrices. These operations are essential for solving a wide range of mathematical problems.

• Addition: Add corresponding elements of two matrices of the same order, \( A + B = C \) where \( c_{ij} = a_{ij} + b_{ij} \).
• Subtraction: Subtract elements of one matrix from the corresponding elements of another matrix of the same size, as seen in the example.
  
• Multiplication: Differs significantly from scalar and matrix subtraction, often involving row-by-column operations.

Each operation requires careful consideration of matrix size and element-wise calculations. These arithmetic operations provide the framework for more complex tasks like solving linear equations and transformations in advanced mathematical contexts.