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Solve the system using the inverse of the coefficient matrix.

   2 x 17 y + 11 z = 0    x + 11 y 7 z = 8                 3 y 2 z = −2

X = [ 4 38 58 ]

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Given a system of equations, solve with matrix inverses using a calculator.

  1. Save the coefficient matrix and the constant matrix as matrix variables [ A ] and [ B ] .
  2. Enter the multiplication into the calculator, calling up each matrix variable as needed.
  3. If the coefficient matrix is invertible, the calculator will present the solution matrix; if the coefficient matrix is not invertible, the calculator will present an error message.

Using a calculator to solve a system of equations with matrix inverses

Solve the system of equations with matrix inverses using a calculator

2 x + 3 y + z = 32 3 x + 3 y + z = −27 2 x + 4 y + z = −2

On the matrix page of the calculator, enter the coefficient matrix    as the matrix variable [ A ] , and enter the constant matrix as the matrix variable [ B ] .

[ A ] = [ 2 3 1 3 3 1 2 4 1 ] , [ B ] = [ 32 −27 −2 ]

On the home screen of the calculator, type in the multiplication to solve for X , calling up each matrix variable as needed.

[ A ] −1 × [ B ]

Evaluate the expression.

[ −59 −34 252 ]
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Access these online resources for additional instruction and practice with solving systems with inverses.

Key equations

Identity matrix for a 2 × 2 matrix I 2 = [ 1 0 0 1 ]
Identity matrix for a 3 × 3 matrix I 3 = [ 1 0 0 0 1 0 0 0 1 ]
Multiplicative inverse of a 2 × 2 matrix A −1 = 1 a d b c [ d b c a ] ,  where  a d b c 0

Key concepts

  • An identity matrix has the property A I = I A = A . See [link] .
  • An invertible matrix has the property A A −1 = A −1 A = I . See [link] .
  • Use matrix multiplication and the identity to find the inverse of a 2 × 2 matrix. See [link] .
  • The multiplicative inverse can be found using a formula. See [link] .
  • Another method of finding the inverse is by augmenting with the identity. See [link] .
  • We can augment a 3 × 3 matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See [link] .
  • Write the system of equations as A X = B , and multiply both sides by the inverse of A : A −1 A X = A −1 B . See [link] and [link] .
  • We can also use a calculator to solve a system of equations with matrix inverses. See [link] .

Section exercises

Verbal

In a previous section, we showed that matrix multiplication is not commutative, that is, A B B A in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, A −1 A = A A −1 ?

If A −1 is the inverse of A , then A A −1 = I , the identity matrix. Since A is also the inverse of A −1 , A −1 A = I . You can also check by proving this for a 2 × 2 matrix.

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Does every 2 × 2 matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

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Can you explain whether a 2 × 2 matrix with an entire row of zeros can have an inverse?

No, because a d and b c are both 0, so a d b c = 0 , which requires us to divide by 0 in the formula.

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Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

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Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 2 × 2 matrix.

Yes. Consider the matrix [ 0 1 1 0 ] . The inverse is found with the following calculation: A −1 = 1 0 ( 0 ) −1 ( 1 ) [ 0 −1 −1 0 ] = [ 0 1 1 0 ] .

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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
Idrissa Reply
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Sherica
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Sherica
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Tamia
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Uday
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salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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