Determine whether the given matrices are inverses of each other. $A=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\text{and}B=\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$

ediculeN
2021-02-04
Answered

Determine whether the given matrices are inverses of each other. $A=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\text{and}B=\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$

You can still ask an expert for help

Derrick

Answered 2021-02-05
Author has **94** answers

Step 1

Here we are given two matrices:

$AB=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$

To show that the given matrices are multiplicative inverses of each other.

Step 2

Multiply AB and BA and if both products equal the identity, then the two matrices are inverses of each other:

Find AB and BA:

$AB=\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]$

$=\left(\begin{array}{ccc}8(-1)+3\cdot 3+(-4)\cdot 0& 8(-1)+3\cdot 4+(-4)\cdot 1& 8(-1)+3\cdot 0+(-4)(-2)\\ (-6)(-1)+(-2)\cdot 3+3\cdot 0& (-6)(-1)+(-2)\cdot 4+3\cdot 1& (-6)(-1)+(-2)\cdot 0+3\cdot (-2)\\ (-3)(-1)+1\cdot 3+1\cdot 0& (-3)(-1)+1\cdot 4+1\cdot 1& (-3)(-1)+1\cdot 0+1\cdot (-2)\end{array}\right)$

$=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 6& 9& 1\end{array}\right)$

Step 3

Find the product BA:

$BA=\left[\begin{array}{ccc}-1& -1& -1\\ 3& 4& 0\\ 0& 1& -2\end{array}\right]\left[\begin{array}{ccc}8& 3& -4\\ -6& -2& 3\\ -3& 1& 1\end{array}\right]$

$=\left[\begin{array}{ccc}(-1)\cdot 8+(-1)(-6)+(-1)(-3)& (-1)\cdot 3+(-1)(-2)+(-1)\cdot 1& (-1)(-4)+(-1)\cdot 3+(-1)1\\ 3\cdot 8+4(-6)+0(-3)& 3\cdot 3+4(-2)+0\cdot 1& 3(-4)+4\cdot 3+0\cdot 1\\ 0\cdot 8+1(-6)+(-2)(-3)& 0\cdot 3+1(-2)+(-2)\cdot 1& 0(-4)+1\cdot 3+(-2)1\end{array}\right]$

$=\left[\begin{array}{ccc}1& -2& 0\\ 0& 1& 0\\ 0& -4& 1\end{array}\right]$

Step 4

So, the product of A and B matrices are not identity matrix. so, that the matrices are not inverse of each other.

Here we are given two matrices:

To show that the given matrices are multiplicative inverses of each other.

Step 2

Multiply AB and BA and if both products equal the identity, then the two matrices are inverses of each other:

Find AB and BA:

Step 3

Find the product BA:

Step 4

So, the product of A and B matrices are not identity matrix. so, that the matrices are not inverse of each other.

Jeffrey Jordon

Answered 2022-01-22
Author has **2027** answers

Answer is given below (on video)

asked 2021-01-31

Find a basis for the space of $2\times 2$ diagonal matrices.

$\text{Basis}=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

asked 2021-02-08

Let B be a 4x4 matrix to which we apply the following operations:

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

asked 2021-01-23

Show that B is the multiplicative inverse of A, where:

$A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]\text{and}B=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

asked 2021-02-13

Solve the system of linear equations using matrices.

x+y+z=3

2x+3y+2z=7

3x-4y+z=4

x+y+z=3

2x+3y+2z=7

3x-4y+z=4

asked 2021-10-24

Construct a $3\times 3$ matrix A, with nonzero entries, and a vector b in $\mathbb{R}}^{3$ such that b is not in the set spanned by the columns of A.

asked 2021-03-18

Explain the term Comparable matrices?

asked 2021-01-31

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A=\left[\begin{array}{cc}-4& 0\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}-2& 4\\ 0& 1\end{array}\right]$