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7.7 Solving systems with inverses  (Page 5/8)

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