Find $BA$, Given:$A=\left(\begin{array}{cc}1 & -3 \ 7 & 2\end{array}\right)$ And $B=\left(\begin{array}{ccc}1 & 0 & -1 \ 3 & 1 & 4\end{array}\right)$.

Introduction

Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices to form a new matrix. Given two matrices A and B, the product AB is defined as the matrix whose entries are the dot products of the rows of A with the columns of B. In this article, we will explore how to find the product BA, given the matrices A and B.

Matrix A and Matrix B

Matrix A is a 2x2 matrix, given by:

$$A=\left(\begin{array}{cc}1 & -3 \\ 7 & 2\end{array}\right)$$

Matrix B is a 2x3 matrix, given by:

$$B=\left(\begin{array}{ccc}1 & 0 & -1 \\ 3 & 1 & 4\end{array}\right)$$

Matrix Multiplication Rules

To find the product BA, we need to follow the rules of matrix multiplication. The number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B). In this case, matrix A has 2 columns and matrix B has 2 rows, so the product BA is possible.

Finding the Product BA

To find the product BA, we need to multiply the rows of matrix B by the columns of matrix A. The resulting matrix will have the same number of rows as matrix B and the same number of columns as matrix A.

Let's calculate the product BA step by step:

• The first row of the product matrix will be the dot product of the first row of matrix B with the columns of matrix A.
• The second row of the product matrix will be the dot product of the second row of matrix B with the columns of matrix A.

Calculating the First Row of the Product Matrix

The first row of the product matrix is the dot product of the first row of matrix B with the columns of matrix A:

$$\left(\begin{array}{c}1 \\ 3\end{array}\right) \cdot \left(\begin{array}{cc}1 & -3 \\ 7 & 2\end{array}\right)$$

Using the dot product formula, we get:

$$\left(\begin{array}{c}1 \\ 3\end{array}\right) \cdot \left(\begin{array}{cc}1 & -3 \\ 7 & 2\end{array}\right) = \left(\begin{array}{c}1 \cdot 1 + 3 \cdot 7 & 1 \cdot (-3) + 3 \cdot 2\end{array}\right)$$

Simplifying the expression, we get:

$$\left(\begin{array}{c}1 \cdot 1 + 3 \cdot 7 & 1 \cdot (-3) + 3 \cdot 2\end{array}\right) = \left(\begin{array}{c}22 & 3\end{array}\right)$$

Calculating the Second Row of the Product Matrix

The second row of the product matrix is the dot product of the second row of matrix B with the columns of matrix A:

$$\left(\begin{array}{c}0 & 1\end{array}\right) \cdot \left(\begin{array}{cc}1 & -3 \\ 7 & 2\end{array}\right)$$

Using the dot product formula, we get:

$$\left(\begin{array}{c}0 & 1\end{array}\right) \cdot \left(\begin{array}{cc}1 & -3 \\ 7 & 2\end{array}\right) = \left(\begin{array}{c}0 \cdot 1 + 1 \cdot 7 & 0 \cdot (-3) + 1 \cdot 2\end{array}\right)$$

Simplifying the expression, we get:

$$\left(\begin{array}{c}0 \cdot 1 + 1 \cdot 7 & 0 \cdot (-3) + 1 \cdot 2\end{array}\right) = \left(\begin{array}{c}7 & 2\end{array}\right)$$

The Product Matrix BA

The product matrix BA is a 2x2 matrix, given by:

$$BA = \left(\begin{array}{cc}22 & 3 \\ 7 & 2\end{array}\right)$$

Conclusion

In this article, we have explored how to find the product BA, given the matrices A and B. We have followed the rules of matrix multiplication and calculated the product matrix step by step. The resulting product matrix BA is a 2x2 matrix, given by:

$$BA = \left(\begin{array}{cc}22 & 3 \\ 7 & 2\end{array}\right)$$

This article has demonstrated the importance of matrix multiplication in linear algebra and has provided a step-by-step guide on how to find the product BA.

Introduction

In our previous article, we explored how to find the product BA, given the matrices A and B. We followed the rules of matrix multiplication and calculated the product matrix step by step. In this article, we will answer some frequently asked questions related to matrix multiplication and provide additional examples to help solidify your understanding.

Q: What are the rules of matrix multiplication?

A: The rules of matrix multiplication are as follows:

• The number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B).
• The resulting matrix will have the same number of rows as matrix B and the same number of columns as matrix A.
• To find the product matrix, we need to multiply the rows of matrix B by the columns of matrix A.

Q: How do I know if matrix multiplication is possible?

A: To determine if matrix multiplication is possible, we need to check if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B). If this condition is met, then matrix multiplication is possible.

Q: What is the difference between AB and BA?

A: The matrices AB and BA are not necessarily equal. In fact, they are only equal if the matrices A and B are square matrices (i.e., they have the same number of rows and columns). If the matrices A and B are not square, then AB and BA will be different.

Q: Can I multiply three or more matrices together?

A: Yes, you can multiply three or more matrices together. However, you need to follow the rules of matrix multiplication carefully. For example, to multiply three matrices A, B, and C, you need to multiply the result of AB by C.

Q: How do I calculate the product of two matrices?

A: To calculate the product of two matrices, you need to follow these steps:

1. Check if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B).
2. If the condition is met, then multiply the rows of matrix B by the columns of matrix A.
3. The resulting matrix will have the same number of rows as matrix B and the same number of columns as matrix A.

Q: What are some common mistakes to avoid when multiplying matrices?

A: Some common mistakes to avoid when multiplying matrices include:

• Not checking if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B).
• Not following the rules of matrix multiplication carefully.
• Not checking if the resulting matrix is square (i.e., it has the same number of rows and columns).

Q: Can I use matrix multiplication to solve systems of linear equations?

A: Yes, you can use matrix multiplication to solve systems of linear equations. In fact, matrix multiplication is a powerful tool for solving systems of linear equations.

Q: How do I use matrix multiplication to solve a system of linear equations?

A: To use matrix multiplication to solve a system of linear equations, you need to follow these steps:

1. Write the system of linear equations in matrix form.
2. Use matrix multiplication to find the product of the coefficient matrix and the variable matrix.
3. Use the resulting matrix to solve for the variables.

Conclusion

In this article, we have answered some frequently asked questions related to matrix multiplication and provided additional examples to help solidify your understanding. We have also discussed some common mistakes to avoid when multiplying matrices and how to use matrix multiplication to solve systems of linear equations. By following the rules of matrix multiplication and being careful when multiplying matrices, you can use matrix multiplication to solve a wide range of problems in linear algebra and beyond.

Additional Examples

Here are some additional examples of matrix multiplication:

Example 1

Find the product of the following matrices:

$$A=\left(\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right)$$

$$B=\left(\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right)$$

To find the product AB, we need to multiply the rows of matrix B by the columns of matrix A:

$$AB = \left(\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right) \cdot \left(\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right)$$

Using the dot product formula, we get:

$$AB = \left(\begin{array}{cc}5 \cdot 1 + 6 \cdot 3 & 5 \cdot 2 + 6 \cdot 4 \\ 7 \cdot 1 + 8 \cdot 3 & 7 \cdot 2 + 8 \cdot 4\end{array}\right)$$

Simplifying the expression, we get:

$$AB = \left(\begin{array}{cc}23 & 34 \\ 31 & 42\end{array}\right)$$

Example 2

Find the product of the following matrices:

$$A=\left(\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right)$$

$$B=\left(\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right)$$

To find the product BA, we need to multiply the rows of matrix A by the columns of matrix B:

$$BA = \left(\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right) \cdot \left(\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right)$$

Using the dot product formula, we get:

$$BA = \left(\begin{array}{cc}1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8\end{array}\right)$$

Simplifying the expression, we get:

$$BA = \left(\begin{array}{cc}19 & 22 \\ 43 & 50\end{array}\right)$$

Example 3

Find the product of the following matrices:

$$A=\left(\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right)$$

$$B=\left(\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right)$$

$$C=\left(\begin{array}{cc}9 & 10 \\ 11 & 12\end{array}\right)$$

To find the product ABC, we need to multiply the result of AB by C:

$$ABC = (AB)C$$

Using the dot product formula, we get:

$$ABC = \left(\begin{array}{cc}23 & 34 \\ 31 & 42\end{array}\right) \cdot \left(\begin{array}{cc}9 & 10 \\ 11 & 12\end{array}\right)$$

Simplifying the expression, we get:

$$ABC = \left(\begin{array}{cc}251 & 284 \\ 313 & 354\end{array}\right)$$