9.7 Solving systems with inverses (Page 2/8)

Precalculus

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Show that the following two matrices are inverses of each other.

A = [\begin{array}{r} 1 & 4 \\ −1 & −3 \end{array}], B = [\begin{array}{r} −3 & −4 \\ 1 & 1 \end{array}]

\begin{array}{l} A B = [\begin{array}{r} 1 & 4 \\ −1 & −3 \end{array}] \begin{array}{r}  \end{array} [\begin{array}{r} −3 & −4 \\ 1 & 1 \end{array}] = [\begin{array}{r} 1 (−3) + 4 (1) & 1 (−4) + 4 (1) \\ −1 (−3) + −3 (1) & −1 (−4) + −3 (1) \end{array}] = [\begin{array}{r} 1 & 0 \\ 0 & 1 \end{array}] \\ B A = [\begin{array}{r} −3 & −4 \\ 1 & 1 \end{array}] \begin{array}{r}  \end{array} [\begin{array}{r} 1 & 4 \\ −1 & −3 \end{array}] = [\begin{array}{r} −3 (1) + −4 (−1) & −3 (4) + −4 (−3) \\ 1 (1) + 1 (−1) & 1 (4) + 1 (−3) \end{array}] = [\begin{array}{r} 1 & 0 \\ 0 & 1 \end{array}] \end{array}

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Finding the multiplicative inverse using matrix multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication .

Finding the multiplicative inverse using matrix multiplication

Use matrix multiplication to find the inverse of the given matrix.

A = [\begin{array}{r} 1 & −2 \\ 2 & −3 \end{array}]

For this method, we multiply $A$ by a matrix containing unknown constants and set it equal to the identity.

[\begin{array}{r} 1 & −2 \\ 2 & −3 \end{array}] [\begin{array}{r} a & b \\ c & d \end{array}] = [\begin{array}{r} 1 & 0 \\ 0 & 1 \end{array}]

Find the product of the two matrices on the left side of the equal sign.

[\begin{array}{r} 1 & −2 \\ 2 & −3 \end{array}] [\begin{array}{r} a & b \\ c & d \end{array}] = [\begin{array}{r} 1 a −2 c & 1 b −2 d \\ 2 a −3 c & 2 b −3 d \end{array}]

Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.

\begin{matrix} 1 a −2 c = 1 R_{1} \\ 2 a −3 c = 0 R_{2} \end{matrix}

Using row operations, multiply and add as follows: $(−2) R_{1} + R_{2} \to R_{2} .$ Add the equations, and solve for $c .$

\begin{array}{r} 1 a - 2 c = 1 \\ 0 + 1 c = - 2 \\ c = - 2 \end{array}

Back-substitute to solve for $a .$

\begin{array}{r} a −2 (−2) = 1 \\ a + 4 = 1 \\ a = −3 \end{array}

Write another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.

\begin{array}{r} 1 b −2 d = 0 & R_{1} \\ 2 b −3 d = 1 & R_{2} \end{array}

Using row operations, multiply and add as follows: $(−2) R_{1} + R_{2} = R_{2} .$ Add the two equations and solve for $d .$

\begin{array}{r} 1 b −2 d = 0 \\ \frac{0 + 1 d = 1}{d = 1} \end{array}

Once more, back-substitute and solve for $b .$

\begin{array}{r} b −2 (1) = 0 \\ b −2 = 0 \\ b = 2 \end{array}

A^{−1} = [\begin{array}{r} −3 & 2 \\ −2 & 1 \end{array}]

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Finding the multiplicative inverse by augmenting with the identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix $A$ is transformed into $I,$ the augmented matrix $I$ transforms into $A^{−1} .$

For example, given

A = [\begin{array}{r} 2 & 1 \\ 5 & 3 \end{array}]

augment $A$ with the identity

[\begin{array}{r} 2 & 1 \\ 5 & 3 \end{array} | \begin{array}{r} 1 & 0 \\ 0 & 1 \end{array}]

Perform row operations with the goal of turning $A$ into the identity.

Switch row 1 and row 2.
$[\begin{array}{r} 5 & 3 \\ 2 & 1 \end{array} | \begin{array}{r} 0 & 1 \\ 1 & 0 \end{array}]$
Multiply row 2 by $−2$ and add to row 1.
$[\begin{array}{r} 1 & 1 \\ 2 & 1 \end{array} | \begin{array}{r} −2 & 1 \\ 1 & 0 \end{array}]$
Multiply row 1 by $−2$ and add to row 2.
$[\begin{array}{r} 1 & 1 \\ 0 & −1 \end{array} | \begin{array}{r} −2 & 1 \\ 5 & −2 \end{array}]$
Add row 2 to row 1.
$[\begin{array}{r} 1 & 0 \\ 0 & −1 \end{array} | \begin{array}{r} 3 & −1 \\ 5 & −2 \end{array}]$
Multiply row 2 by $−1.$
$[\begin{array}{r} 1 & 0 \\ 0 & 1 \end{array} | \begin{array}{r} 3 & −1 \\ −5 & 2 \end{array}]$

The matrix we have found is $A^{−1} .$

A^{−1} = [\begin{array}{r} 3 & −1 \\ −5 & 2 \end{array}]

Finding the multiplicative inverse of 2×2 matrices using a formula

When we need to find the multiplicative inverse of a $2 \times 2$ matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.

If $A$ is a $2 \times 2$ matrix, such as

A = [\begin{array}{r} a & b \\ c & d \end{array}]

the multiplicative inverse of $A$ is given by the formula

A^{−1} = \frac{1}{a d - b c} [\begin{array}{r} d & - b \\ - c & a \end{array}]

where $a d - b c \neq 0.$ If $a d - b c = 0,$ then $A$ has no inverse.

Using the formula to find the multiplicative inverse of matrix A

Use the formula to find the multiplicative inverse of

A = [\begin{matrix} 1 & −2 \\ 2 & −3 \end{matrix}]

Using the formula, we have

\begin{array}{l} A^{−1} = \frac{1}{(1) (−3) - (−2) (2)} [\begin{matrix} −3 & 2 \\ −2 & 1 \end{matrix}] \\ = \frac{1}{−3 + 4} [\begin{matrix} −3 & 2 \\ −2 & 1 \end{matrix}] \\ = [\begin{matrix} −3 & 2 \\ −2 & 1 \end{matrix}] \end{array}

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?

ADNAN Reply

linear speed of an object

Melissa Reply

an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object

Melissa

test

Matrix

how to find domain

Mohamed Reply

like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.

Dan

define the term of domain

Moha

if a>0 then the graph is concave

Angel Reply

if a<0 then the graph is concave blank

Angel

what's a domain

Kamogelo Reply

The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.

Spiro

how fast can i understand functions without much difficulty

Joe Reply

what is inequalities

Nathaniel

functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2

Dan

what is set?

Kelvin Reply

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size

Divya Reply

I got 300 minutes. is it right?

Patience

no. should be about 150 minutes.

Jason

It should be 158.5 minutes.

ok, thanks

Patience

100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes

Thomas

158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49

Dan

what is the importance knowing the graph of circular functions?

Arabella Reply

can get some help basic precalculus

ismail Reply

What do you need help with?

Andrew

how to convert general to standard form with not perfect trinomial

Camalia Reply

can get some help inverse function

ismail

Rectangle coordinate

Asma Reply

how to find for x

Jhon Reply

it depends on the equation

Robert

yeah, it does. why do we attempt to gain all of them one side or the other?

Melissa

how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2

Dan

whats a domain

mike Reply

The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.

Spiro

Spiro; thanks for putting it out there like that, 😁

Melissa

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.

Churlene Reply

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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