Problem 45 - Algebra and Trigonometry

Question

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Find the product of the sum of A and B and the difference between C and D.

Step-by-Step Solution

Answer

The product of the sum of matrix A and B and the difference between matrix C and D is a 2x2 zero matrix. \[\left[\begin{array}{cc} 0 & 0 \ 0 & 0\end{array}\right]\].

Step 1: Compute the Sum of Matrices A and B

Adding two matrices involves adding the corresponding elements. The sum of A and B, denoted as A+B, is a new matrix whose element in the i-th row and j-th column can be calculated as: \((A+B)_{ij}=A_{ij}+B_{ij}\). Applying this method gives the following matrix: \[ A+B=\left[\begin{array}{cc} 1+1 & 0+0 \ 0+0 & 1+(-1) \end{array}\right]=\left[\begin{array}{cc} 2 & 0 \ 0 & 0 \end{array}\right]. \]

Step 2: Compute the Difference between Matrices C and D

Subtracting two matrices involves subtracting the corresponding elements. The difference of C and D, denoted as C-D, is a new matrix whose element in the i-th row and j-th column can be calculated as: \((C-D)_{ij}=C_{ij}-D{ij}\). Applying this method gives the following matrix: \[ C-D=\left[\begin{array}{cc} -1-(-1) & 0-0 \ 0-0 & 1-(-1) \end{array}\right]=\left[\begin{array}{cc} 0 & 0 \ 0 & 2 \end{array}\right]. \]

Step 3: Compute the Product of Matrices (A+B) and (C-D)

Multiplication of two matrices results in a new matrix in which each element is the dot product of the corresponding row of the first matrix and the column of the second matrix. Let’s multiply (A+B) and (C-D) to get our final output: \[(A+B) \times (C-D)=\left[\begin{array}{cc} 2 & 0 \ 0 & 0 \end{array}\right] \times \left[\begin{array}{cc} 0 & 0 \ 0 & 2 \end{array}\right]=\left[\begin{array}{cc} 0 & 0 \ 0 & 0 \end{array}\right].\]

Key Concepts

Matrix Addition

Matrix addition is a straightforward concept that involves adding corresponding elements from two different matrices. Imagine each matrix like a grid, where numbers sit in boxes lined up in neat rows and columns. To add them up, you simply go box by box, adding the numbers from the same position in each matrix.

• For this operation to work, the matrices must have the same dimensions. This means they should have the same number of rows and columns.
• Each element in the resulting matrix is the sum of the elements from the two matrices at the same position.

In the exercise, the sum of two given matrices, A and B, results in a new matrix:\[A+B=\left[\begin{array}{cc} 2 & 0 \ 0 & 0 \end{array}\right].\]This adds the corresponding numbers from A and B to produce a new set of values, element by element. With matrix addition, it's all about aligning numbers to add them together efficiently.

Matrix Subtraction

Matrix subtraction works much like matrix addition, but instead of adding, you subtract the matching elements of one matrix from those in the other.

• Here, again, the matrices must have the same dimensions for the operation to be valid.
• Each element in the resulting matrix is the difference between the elements from the two matrices at the same position.

When we look at matrices C and D in the exercise, their subtraction is calculated as:\[C-D=\left[\begin{array}{cc} 0 & 0 \ 0 & 2 \end{array}\right].\]This involves subtracting each element in matrix D from the corresponding element in matrix C. Matrix subtraction is another essential arithmetic operation that allows for precise mathematical calculations between matrices.

Identity Matrix

An identity matrix is a special type of matrix that acts as the "1" in matrix multiplication, meaning that any matrix multiplied by the identity matrix remains unchanged. It's a square matrix, featuring ones on the main diagonal from the top left to the bottom right and zeros everywhere else.

• For a 2x2 identity matrix, it looks like this: \[I=\left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right].\]
• In multiplication, any matrix, say matrix A, multiplied by the identity matrix, I, will be A again: \[A \times I = A.\]

The concept of the identity matrix is crucial in understanding more complex operations like matrix inversion and solving matrix equations. It serves as a fundamental building block in linear algebra and ensures that the essence or structure of a matrix remains intact during multiplication.