Problem 14 - 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. $$ A-4 I_{2} $$

Step-by-Step Solution

Answer

The result of \(A - 4I_{2}\) is \(\begin{bmatrix} -3 & 2 \\ 3 & 0 \end{bmatrix}\).

Step 1: Identify Matrix Dimensions

Matrix \(A\) is a 2x2 matrix, and \(I_{2}\) represents the 2x2 identity matrix, \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). Both matrices have the same dimensions, so the operations can be performed.

Step 2: Scalar Multiplication of Identity Matrix

Multiply the identity matrix \(I_{2}\) by 4: \[4 I_{2} = 4 \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix}\]

Step 3: Subtract Scaled Matrix from A

Subtract the resulting scaled identity matrix from matrix \(A\):\[A - 4 I_{2} = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} - \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} = \begin{bmatrix} 1-4 & 2-0 \ 3-0 & 4-4 \end{bmatrix} = \begin{bmatrix} -3 & 2 \ 3 & 0 \end{bmatrix}\]

Key Concepts

Subtraction of Matrices

Matrix subtraction is straightforward, but it's crucial to ensure that both matrices have the same dimensions. This requirement is because matrix subtraction involves subtracting corresponding elements. For example, if you have two 2x2 matrices, you subtract the element in the first row and first column of one matrix from the element in the first row and first column of the other matrix.
Always perform the subtraction component-wise:

• For matrices like `[a]` in matrix A and `[b]` in matrix B, the subtraction becomes `[a - b]`.
• Ensure to maintain the order: The matrix from which you are subtracting is always listed first.
• If the matrices have different dimensions, the operation is undefined.

Scalar Multiplication

Scalar multiplication involves multiplying each entry of the matrix by the same scalar value. It is essential when you want to scale up a matrix by a constant factor. For example, if you have a scalar 4 and a 2x2 matrix, multiply every element in the matrix by 4. This operation helps in situations where you need to adjust the magnitude of a matrix's elements without changing their layout.

Consider multiplying an identity matrix by a scalar:

• Scalar 4 multiplying identity matrix: \[ 4 \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} \]
• Each element in the matrix absorbs the scalar value, spreading it evenly across its dimensions.

Identity Matrix

The identity matrix is a special type of square matrix with ones on the main diagonal and zeros elsewhere. In matrix operations, it acts like the number 1 does in multiplication for numbers. When you multiply any matrix by the identity matrix, you end up with the original matrix itself. The identity matrix is denoted by \(I\), and in this context, we're exploring the 2x2 identity matrix, which looks like: \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

• The main diagonal contains only ones, confirming its identity property.
• When using identity matrices in subtraction, it behaves uniquely, due to the zeros in non-diagonal places.
• Its primary function in subtraction or addition is maintaining the matrix structure, while applying effects of scalar multiplication efficiently.

2x2 Matrices

These matrices are among the simplest forms of matrices but are fundamental in understanding larger matrix operations. A 2x2 matrix comprises two rows and two columns, creating a small grid of values: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
They are commonly used in a variety of mathematical computations, from solving systems of equations to performing transformations in graphics. The beauty of 2x2 matrices lies in their simplicity and manageability when grasping core matrix concepts.

• Handling operations such as addition, subtraction, and multiplication becomes easier when starting with 2x2 matrices.
• They serve as a perfect starting point to practice matrix operations due to their consistent format and ease of understanding the placement of elements.

2x2 matrices in exercises often help reinforce the idea that matrices must be of the same size to perform certain operations, easing students into these foundational rules.