Problem 27 - College Algebra

Question

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{ll}1 & 3 \\\5 & 3\end{array}\right], \quad B=\left[\begin{array}{rr}3 & -2 \\\\-1 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The product AB is a=[[0,16],[12,8]] and the product BA is b=[[-7,3],[29,15]].

Step 1 Calculate the product A times B

According to the rule of matrix multiplication, the element at the ith row and jth column of the resulting matrix is the dot product of the ith row of the first matrix and the jth column of the second matrix. Hence the multiplication A times B results in the following first row [1*3 + 3*(-1), 1*(-2) + 3 * 6]=[0, 16] and the second row results in [5*3 + 3*(-1), 5*(-2) + 3 * 6] = [12, 8]. So matrix AB, denoted as a, is [[0,16],[12,8]].

Step 2 Calculate the product B times A

Just as before we perform the matrix multiplication according to the rules. First row will be [3*1 + (-2)*5, 3*3 + (-2)*3]=[-7,3], and second row will be [-(1*1) + 6*5, -(1*3) + 6*3] = [29,15]. Hence the matrix BA, denoted as b, is [[-7,3],[29,15]].

Key Concepts

Matrix Product

When we talk about the matrix product, we refer to the result of multiplying two matrices, A and B. This operation is a fundamental concept in linear algebra and differs from regular multiplication numbers. It involves a step-by-step process where you carry out dot products of rows from one matrix with columns from the next.

The key things to remember when multiplying matrices include:

• Matrix multiplication isn't commutative, meaning \( A \times B eq B \times A \) in general.
• Each element of the resulting matrix is obtained by multiplying corresponding elements and adding them.
• The order of matrices matters for this operation's validity, which leads us to understand matrices order next.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations like matrix operations. Understanding linear algebra is vital because it provides the tools to handle systems of linear equations, perform transformations, and solve practical problems in engineering, physics, computer science, and more.

At its core, linear algebra involves:

• Vectors and matrix representations.
• Matrix operations such as addition, subtraction, and multiplication.
• Solving linear systems using techniques like Gaussian elimination.
• Eigenvalues and eigenvectors, which reveal important properties of matrices.

When dealing with matrices, understanding the background of linear algebra helps in grasping concepts like matrix product and matrices order.

Matrices Order

The order of matrices is crucial for determining which operations can be performed. The order is given in terms of the number of rows and columns that a matrix has, which is denoted as \( m \times n \) (with \( m \) being the number of rows and \( n \) being the number of columns).

Some important notes on matrix order:

• For two matrices to be multiplied, the number of columns in the first must equal the number of rows in the second. For example, if matrix \( A \) is of order \( 2 \times 3 \), matrix \( B \) must be of order \( 3 \times n \) for the product \( A \times B \) to be defined.
• The product matrix will have the order given by the number of rows of the first matrix and the number of columns of the second.
• In our example, \( A \) is \( 2 \times 2 \) and \( B \) is \( 2 \times 2 \). Thus, both \( A \times B \) and \( B \times A \) are possible, resulting in \( 2 \times 2 \) matrices.

Matrix Operations

Matrix operations are varied and provide meaningful manipulation of data represented in matrix form. They include more than just the matrix product; common operations include addition, subtraction, and scalar multiplication. Each operation follows specific rules of linear algebra:

• **Addition and Subtraction**: Only matrices of the same order can be added or subtracted. The elements in corresponding positions are added or subtracted.
• **Scalar Multiplication**: Every element of the matrix is multiplied by a number or scalar. This operation is simple yet fundamental in rescaling matrix data.

These operations allow matrices to be used in various applications, from solving linear systems to performing transformations in graphics and data analysis.