Problem 10 - Algebra and Trigonometry

Question

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rr} {-2} & {3} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{ll} {8} & {1} \\ {5} & {4} \end{array}\right] $$

Step-by-Step Solution

Answer

\(A+B = \left[\begin{array}{cc} 6 & 4 \ 5 & 5 \end{array}\right]\), \(A-B = \left[\begin{array}{cc} -10 & 2 \ -5 & -3 \end{array}\right]\), \(-4A=\left[\begin{array}{cc} 8 & -12 \ 0 & -4 \end{array}\right]\), \(3A+2B = \left[\begin{array}{cc} 4 & 11 \ 10 & 11 \end{array}\right]\)

Step 1: Matrix Addition

For matrix addition (or subtraction), just add (or subtract) the corresponding entries from each matrix. For \(A + B\), \[ \left[\begin{array}{rr}-2 & 3 \ 0 & 1\end{array}\right] + \left[\begin{array}{ll}8 & 1 \ 5 & 4\end{array}\right] = \left[\begin{array}{cc} (-2+8) & (3+1) \ (0+5) & (1+4)\end{array}\right] = \left[\begin{array}{cc} 6 & 4 \ 5 & 5 \end{array}\right] \]

Step 2: Matrix Subtraction

For \(A - B\), \[ \left[\begin{array}{rr}-2 & 3 \ 0 & 1\end{array}\right] - \left[\begin{array}{ll}8 & 1 \ 5 & 4\end{array}\right] = \left[\begin{array}{cc} (-2-8) & (3-1) \ (0-5) & (1-4)\end{array}\right] = \left[\begin{array}{cc} -10 & 2 \ -5 & -3 \end{array}\right] \]

Step 3: Scalar Multiplication

A scalar (\(-4\)) times a matrix (A) means we multiply each entry of the matrix by that scalar. So, \(-4A = -4 \times \left[\begin{array}{rr}-2 & 3 \ 0 & 1\end{array}\right] = \left[\begin{array}{cc} (-4 \times -2) & (-4 \times 3) \ (-4 \times 0) & (-4 \times 1) \end{array}\right] = \left[\begin{array}{cc} 8 & -12 \ 0 & -4 \end{array}\right] \]

Step 4: Combination of Operations

For \(3A + 2B\), perform scalar multiplication first, then add up the two resulting matrices. \[3 \times \left[\begin{array}{rr} -2 & 3 \ 0 & 1\end{array}\right] + 2 \times \left[\begin{array}{ll} 8 & 1 \ 5 & 4\end{array}\right] = \left[\begin{array}{cc} (3 \times -2)+ (2 \times 8) & (3 \times 3) + (2 \times 1) \ (3 \times 0)+ (2 \times 5) & (3 \times 1) + (2 \times 4) \end{array}\right] = \left[\begin{array}{cc} 4 & 11 \ 10 & 11 \end{array}\right] \]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It is used when you want to combine the information contained in two matrices. To add two matrices, they must have the same dimensions. This means the same number of rows and columns.

When adding matrices, you combine their corresponding elements. For example, if you have two 2x2 matrices, you add the element in the first row and first column of the first matrix to the element in the first row and first column of the second matrix, and so forth.

• For matrix A, element \(a_{1,1}\) is added to element \(b_{1,1}\) of matrix B.
• The operation continues for each element across the entire matrix structure.

Thus, the addition of matrices A and B follows this format: \[ A + B = \begin{bmatrix}(a_{1,1} + b_{1,1}) & (a_{1,2} + b_{1,2}) \ (a_{2,1} + b_{2,1}) & (a_{2,2} + b_{2,2})\end{bmatrix} \]This operation results in a new matrix containing the sums of the corresponding elements.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition, but instead of adding corresponding elements, you subtract them. Just like addition, both matrices must have the same dimensions. This ensures that each element from one matrix has a corresponding element in the other matrix to subtract from.

When subtracting matrices, perform the operation element-wise:

• Take the first element of the first matrix and subtract the first element of the second matrix, and repeat this for each corresponding pair of elements.

Thus, for matrices A and B, matrix subtraction can be represented as:\[ A - B = \begin{bmatrix}(a_{1,1} - b_{1,1}) & (a_{1,2} - b_{1,2}) \ (a_{2,1} - b_{2,1}) & (a_{2,2} - b_{2,2})\end{bmatrix} \]By following this element-wise operation, you'll obtain a new matrix representing the differences of the corresponding elements. This step is crucially important in matrix arithmetic for computations such as finding differences in data sets.

Scalar Multiplication

Scalar multiplication in matrices involves multiplying every element within a matrix by a single number, which is known as a scalar. This process is straightforward and is like scaling the matrix by expanding or contracting it by the scalar value.

To perform scalar multiplication:

• Each individual element of the matrix is multiplied by the scalar.
• This operation is conducted independently for each element.

Let's take an example: If you have a matrix \(A\) and a scalar \(k\), the operation is:\[ kA = k \times \begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix} = \begin{bmatrix}k \times a_{11} & k \times a_{12} \k \times a_{21} & k \times a_{22}\end{bmatrix} \]This operation is very useful in many mathematical and real-world applications, such as altering proportions in matrices that represent scaled systems like economics or physics.

Matrix Arithmetic

Matrix arithmetic encompasses several operations including addition, subtraction, and multiplication by scalars. These operations are the building blocks of more complex computations like solving systems of equations, transformations, and analyzing statistical data. Understanding each of these components individually allows for mastering complex mathematical tasks that require manipulation and computation on matrices.

To effectively perform matrix arithmetic:

• Ensure matrices have compatible dimensions for operations.
• For addition and subtraction, element-wise combine the matrices.
• For scalar multiplication, multiply each matrix element by the scalar value.
• Recognize that commutative and associative properties apply for addition, but not necessarily for multiplication.

In practical and theoretical applications, these operations help transition raw data into comprehensible formats, assisting in analytical assessments and model building. By understanding matrix arithmetic, you gain tools to solve real-world problems through structured, step-by-step mathematical modeling.