Select All The Correct Answers.In Which Pairs Of Matrices Does A B = B A A B = B A A B = B A ?1. A = [ 1 0 − 2 1 ] A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix} A = [ 1 − 2 ​ 0 1 ​ ] , $B=\begin{bmatrix} 5 & 0 \ 3 & 2 \end{bmatrix}$2. $A=\begin{bmatrix} 1 & 0 \ -1 & 2

In linear algebra, matrix multiplication is a fundamental operation that combines two matrices to produce another matrix. However, unlike scalar multiplication, matrix multiplication is not always commutative, meaning that the order of the matrices matters. In this article, we will explore the conditions under which matrix multiplication is commutative, and we will examine specific pairs of matrices to determine if $A B = B A$.

Commutative Property of Matrix Multiplication

The commutative property of matrix multiplication states that if $A$ and $B$ are two matrices, then $A B = B A$ if and only if $A B = B A$. This means that the order of the matrices does not affect the result of the multiplication. However, this property does not hold in general, and there are many cases where $A B \neq B A$.

Conditions for Commutative Matrix Multiplication

For matrix multiplication to be commutative, the following conditions must be satisfied:

1. Matrix dimensions: The number of columns in the first matrix $A$ must be equal to the number of rows in the second matrix $B$.
2. Matrix dimensions: The number of rows in the first matrix $A$ must be equal to the number of columns in the second matrix $B$.
3. Matrix elements: The elements of the matrices must be such that the multiplication is commutative.

Example 1: $A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}$, $B=\begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix}$

Let's examine the first pair of matrices:

$$A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}, B=\begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix}$$

To determine if $A B = B A$, we need to calculate both products.

Calculating $A B$

$$A B = \begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix}$$

$$A B = \begin{bmatrix} 1 \cdot 5 + 0 \cdot 3 & 1 \cdot 0 + 0 \cdot 2 \\ -2 \cdot 5 + 1 \cdot 3 & -2 \cdot 0 + 1 \cdot 2 \end{bmatrix}$$

$$A B = \begin{bmatrix} 5 & 0 \\ -7 & 2 \end{bmatrix}$$

Calculating $B A$

$$B A = \begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}$$

$$B A = \begin{bmatrix} 5 \cdot 1 + 0 \cdot (-2) & 5 \cdot 0 + 0 \cdot 1 \\ 3 \cdot 1 + 2 \cdot (-2) & 3 \cdot 0 + 2 \cdot 1 \end{bmatrix}$$

$$B A = \begin{bmatrix} 5 & 0 \\ 1 & 2 \end{bmatrix}$$

Since $A B \neq B A$, the commutative property of matrix multiplication does not hold for this pair of matrices.

Example 2: $A=\begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}$, $B=\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$

Let's examine the second pair of matrices:

$$A=\begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}, B=\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$$

To determine if $A B = B A$, we need to calculate both products.

Calculating $A B$

$$A B = \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$$

$$A B = \begin{bmatrix} 1 \cdot 2 + 0 \cdot 1 & 1 \cdot 0 + 0 \cdot 3 \\ -1 \cdot 2 + 2 \cdot 1 & -1 \cdot 0 + 2 \cdot 3 \end{bmatrix}$$

$$A B = \begin{bmatrix} 2 & 0 \\ 0 & 6 \end{bmatrix}$$

Calculating $B A$

$$B A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}$$

$$B A = \begin{bmatrix} 2 \cdot 1 + 0 \cdot (-1) & 2 \cdot 0 + 0 \cdot 2 \\ 1 \cdot 1 + 3 \cdot (-1) & 1 \cdot 0 + 3 \cdot 2 \end{bmatrix}$$

$$B A = \begin{bmatrix} 2 & 0 \\ -2 & 6 \end{bmatrix}$$

Since $A B \neq B A$, the commutative property of matrix multiplication does not hold for this pair of matrices.

Conclusion

In conclusion, matrix multiplication is not always commutative, and the order of the matrices matters. We have examined two pairs of matrices and found that the commutative property of matrix multiplication does not hold for either pair. The conditions for commutative matrix multiplication are strict, and not all pairs of matrices satisfy these conditions. Therefore, it is essential to carefully examine the dimensions and elements of the matrices before performing matrix multiplication.

Final Answer

Based on the calculations, we can conclude that:

• For the pair of matrices $A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}$, $B=\begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix}$, $A B \neq B A$.
• For the pair of matrices $A=\begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}$, $B=\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$, $A B \neq B A$.

In this article, we will address some common questions related to matrix multiplication and the commutative property.

Q: What is the commutative property of matrix multiplication?

A: The commutative property of matrix multiplication states that if $A$ and $B$ are two matrices, then $A B = B A$ if and only if $A B = B A$. This means that the order of the matrices does not affect the result of the multiplication.

Q: When does the commutative property of matrix multiplication hold?

A: The commutative property of matrix multiplication holds when the number of columns in the first matrix $A$ is equal to the number of rows in the second matrix $B$, and the number of rows in the first matrix $A$ is equal to the number of columns in the second matrix $B$.

Q: What are the conditions for commutative matrix multiplication?

A: The conditions for commutative matrix multiplication are:

1. Matrix dimensions: The number of columns in the first matrix $A$ must be equal to the number of rows in the second matrix $B$.
2. Matrix dimensions: The number of rows in the first matrix $A$ must be equal to the number of columns in the second matrix $B$.
3. Matrix elements: The elements of the matrices must be such that the multiplication is commutative.

Q: Can you provide an example of matrices that satisfy the commutative property of matrix multiplication?

A: Yes, here is an example of matrices that satisfy the commutative property of matrix multiplication:

$$A=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

In this case, $A B = B A$.

Q: Can you provide an example of matrices that do not satisfy the commutative property of matrix multiplication?

A: Yes, here is an example of matrices that do not satisfy the commutative property of matrix multiplication:

$$A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}, B=\begin{bmatrix} 5 & 0 \\ 3 & 2 \end{bmatrix}$$

In this case, $A B \neq B A$.

Q: What is the significance of the commutative property of matrix multiplication?

A: The commutative property of matrix multiplication is significant because it allows us to simplify matrix equations and perform matrix operations more efficiently. However, it is essential to note that not all pairs of matrices satisfy the commutative property, and we must carefully examine the dimensions and elements of the matrices before performing matrix multiplication.

Q: Can you provide a formula for calculating the product of two matrices?

A: Yes, the formula for calculating the product of two matrices $A$ and $B$ is:

$$A B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22} \end{bmatrix}$$

Q: Can you provide a formula for calculating the product of two matrices using the dot product?

A: Yes, the formula for calculating the product of two matrices $A$ and $B$ using the dot product is:

$$A B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} \cdot b_{11} + a_{12} \cdot b_{21} & a_{11} \cdot b_{12} + a_{12} \cdot b_{22} \\ a_{21} \cdot b_{11} + a_{22} \cdot b_{21} & a_{21} \cdot b_{12} + a_{22} \cdot b_{22} \end{bmatrix}$$

Q: Can you provide a formula for calculating the product of two matrices using the matrix multiplication algorithm?

A: Yes, the formula for calculating the product of two matrices $A$ and $B$ using the matrix multiplication algorithm is:

1. Initialize an empty matrix $C$ with the same number of rows as $A$ and the same number of columns as $B$.
2. For each element $c_{ij}$ in the matrix $C$, calculate the dot product of the $i$-th row of $A$ and the $j$-th column of $B$.
3. Store the result of the dot product in the matrix $C$.

Conclusion

In conclusion, matrix multiplication is a fundamental operation in linear algebra, and the commutative property of matrix multiplication is a crucial concept to understand. We have addressed some common questions related to matrix multiplication and provided examples of matrices that satisfy and do not satisfy the commutative property. We have also provided formulas for calculating the product of two matrices using the dot product and the matrix multiplication algorithm.