Problem 8 - Algebra 2

Question

Use matrices \(A, B, C,\) and \(D .\) Find each product, sum, or difference. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{ll}{5} & {1} \\ {0} & {2}\end{array}\right]$$ $$ 2 A-5 B $$

Step-by-Step Solution

Answer

\left[\begin{array}{rr}21 & 3 \ 2 & 16 \ 7 & -25\end{array}\right]

Step 1: Understand Matrix Multiplication and Addition

To find a product of matrices, we multiply corresponding entries and sum them to form a new matrix. Matrix addition and subtraction are performed by adding or subtracting corresponding entries of the matrices.

Step 2: Multiply Scalar 2 with Matrix A

To find 2A, multiply each entry of matrix A by 2.\[2A = 2\cdot\left[\begin{array}{rr}3 & 4 \ 6 & -2 \ 1 & 0\end{array}\right] = \left[\begin{array}{rr}6 & 8 \ 12 & -4 \ 2 & 0\end{array}\right]\]

Step 3: Multiply Scalar -5 with Matrix B

To find -5B, multiply each entry of matrix B by -5.\[-5B = -5\cdot\left[\begin{array}{rr}-3 & 1 \ 2 & -4 \ -1 & 5\end{array}\right] = \left[\begin{array}{rr}15 & -5 \ -10 & 20 \ 5 & -25\end{array}\right]\]

Step 4: Find the Sum 2A - 5B

Add the matrices 2A and -5B by summing their corresponding entries.\[2A-5B = \left[\begin{array}{rr}6 & 8 \ 12 & -4 \ 2 & 0\end{array}\right] + \left[\begin{array}{rr}15 & -5 \ -10 & 20 \ 5 & -25\end{array}\right] = \left[\begin{array}{rr}6+15 & 8+(-5) \ 12+(-10) & -4+20 \ 2+5 & 0+(-25)\end{array}\right]\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra that involves combining two matrices by adding their corresponding entries. This operation can only be performed on matrices of the same dimension, meaning that the matrices must have the same number of rows and columns. When adding matrices, we simply take each element from the first matrix and add it to the corresponding element in the second matrix.

For example, if we have two matrices, Matrix A and Matrix B, their sum, denoted by \(A + B\), would be computed as follows:

• Add the entry in the first row and first column of Matrix A to the entry in the first row and first column of Matrix B.
• Continue this process for each corresponding entry.

Once all entries have been added, the resulting matrix will be the same size as the original matrices and each element of the resulting matrix will be the sum of the corresponding elements from the two matrices added together.

Scalar Multiplication in Matrices

Scalar multiplication in matrices involves multiplying each entry of a matrix by a single number, known as a scalar. This operation affects every element of the matrix and is essential in various matrix algebra calculations and applications.

To perform scalar multiplication, we apply the scalar to each entry independently. The process is straightforward:

• Take scalar \(k\) and matrix \( M \).
• Multiply each entry \(m_{ij}\) of matrix \( M \) by scalar \(k\) to produce a new matrix \(kM\), where the \(i\)-th row and \(j\)-th column entry is \( km_{ij} \).

For instance, if we multiply Matrix A by scalar 2, denoted as \(2A\), we double each element within the matrix A. This operation is useful when scaling figures or adjusting values across entire data sets in matrices.

Matrix Algebra

Matrix algebra encompasses a variety of operations including addition, scalar multiplication, and more complex actions such as matrix multiplication and finding determinants or inverses. It forms a cornerstone of linear algebra and is widely applied across numerous fields including computer science, physics, and engineering.

Matrix algebra operates under specific rules that are somewhat similar to but also distinct from, arithmetic with numbers. For example, while matrix addition is commutative (i.e., \(A + B = B + A\)), matrix multiplication is not necessarily commutative. Understanding these operations and their properties is crucial for utilizing matrices to solve systems of equations, perform geometric transformations, and analyze large data sets effectively.

Returning to our problem, we already performed matrix addition and scalar multiplication. To solve other matrix-related exercises, one might encounter tasks requiring the application of matrix multiplication (aligning and multiplying rows by columns), finding inverses (for non-singular, square matrices), or determining eigenvalues and eigenvectors (which reveal important properties of matrices). Each of these operations relies on the foundational concepts of matrix addition and scalar multiplication, demonstrating the interconnected nature of matrix algebra.