Problem 12 - Algebra for College Students

Question

For Problems \(1-12\), compute the indicated matrix by using the following matrices: \(\begin{array}{ll} A=\left[\begin{array}{rr} 1 & -2 \\ 3 & 4 \end{array}\right] & B=\left[\begin{array}{lr} 2 & -3 \\ 5 & -1 \end{array}\right] \\ C=\left[\begin{array}{rr} 0 & 6 \\ -4 & 2 \end{array}\right] & D=\left[\begin{array}{rr} -2 & 3 \\ 5 & -4 \end{array}\right] \\ E=\left[\begin{array}{lr} 2 & 5 \\ 7 & 3 \end{array}\right] & \end{array}\) $$ A-(B+C) $$

Step-by-Step Solution

Answer

The result of A - (B + C) is \( \begin{bmatrix} -1 & -5 \\ 2 & 3 \end{bmatrix} \).

Step 1: Calculate B + C

First, calculate the sum of matrices B and C. Matrix B is \( \begin{bmatrix} 2 & -3 \ 5 & -1 \end{bmatrix} \) and matrix C is \( \begin{bmatrix} 0 & 6 \ -4 & 2 \end{bmatrix} \). Add corresponding elements from each matrix: \( B + C = \begin{bmatrix} 2+0 & -3+6 \ 5+(-4) & -1+2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \ 1 & 1 \end{bmatrix} \)

Step 2: Calculate A - (B + C)

Now, subtract the matrix sum \( B + C \) from matrix A. Matrix A is \( \begin{bmatrix} 1 & -2 \ 3 & 4 \end{bmatrix} \). Subtract the corresponding elements of the matrices: \( A - (B + C) = \begin{bmatrix} 1-2 & -2-3 \ 3-1 & 4-1 \end{bmatrix} = \begin{bmatrix} -1 & -5 \ 2 & 3 \end{bmatrix} \)

Key Concepts

Matrix Addition

Matrix addition is one of the basic operations in linear algebra, and it involves adding together matrices of the same size. Given two matrices, their sum is a matrix where each element is the sum of the corresponding elements from the original matrices. For example, if you have matrix

• Matrix A with elements a_ij
• Matrix B with elements b_ij

Then, the resulting matrix after addition will be matrix C, having elements c_{ij = aij + bij.
Adding matrices is straightforward: just ensure the matrices are of the same dimensions. In the exercise's first step, you added matrices B and C to get a new matrix.
Always remember, you can't add matrices with different dimensions, like trying to add a 2x2 matrix with a 3x3 matrix. The number of rows and columns must match for the addition to be possible.}

Matrix Subtraction

Matrix subtraction is similar to matrix addition. It involves subtracting corresponding elements of matrices. Like addition, subtraction requires the matrices to be of the same size.
When we subtract matrix B from matrix A, the resulting matrix has each element defined as:

• c_ij
= a_{ij - bij}

Where a_{ij are elements of matrix A and bij are elements of matrix B.
In the original exercise, after adding matrices B and C, you subtract their sum from matrix A. Subtraction works just like addition: correspond each element with each other, then perform the subtraction.
One thing to always ensure is confirming the dimensions match. Subtraction won't be defined for matrices of different dimensions, just like in addition.}

Matrix Arithmetic

Matrix arithmetic refers to performing operations such as addition, subtraction, and other calculations involving matrices. These operations form the foundation of linear algebra and allow solving complex problems with ease.
These calculations involve rule sets, just like regular arithmetic, but with added complexity of simultaneously dealing with rows and columns.
After learning addition through the example provided, you might also come across other matrix-based calculations such as scalar multiplication, where a matrix is multiplied by a single number to scale its elements.

• Addition and subtraction involve coordinating element-wise operations.
• Ensure all matrices involved in addition, subtraction, or multiplication have compatible dimensions.

Getting comfortable with these basic operations is crucial before delving into more complex topics like matrix multiplication and inversion.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between them. It includes matrix operations such as addition and subtraction but also stretches to more advanced concepts.
It is widely applicable in various scientific areas, including physics, computer science, and engineering. Linear algebra provides the language for solving systems of linear equations, performing transformations, and understanding intrinsic properties of mathematical constructions.
The exercise given is part of the greater study of linear algebra, showcasing basic matrix arithmetic's importance in problem-solving. As you move forward:

• You'll encounter vector spaces, determinants, eigenvalues, and eigenvectors.
• Each concept builds from foundational knowledge of matrix arithmetic.

Understanding these basics paves the way for tackling complex problems and truly appreciating linear algebra's role in advanced mathematical methodologies.