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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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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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