Page 152 - 35Linear Algebra

152 Matrices

To solve many linear systems with the same matrix at once,

MX = V 1 , MX = V 2

we can consider augmented matrices with many columns on the right and
then apply Gaussian row reduction to the left side of the matrix. Once the
identity matrix is on the left side of the augmented matrix, then the solution
of each of the individual linear systems is on the right.

−1 −1
M V 1 V 2 ∼ I M V 1 M V 2

To compute M −1 , we would like M −1 , rather than M −1 V to appear on
the right side of our augmented matrix. This is achieved by solving the
collection of systems MX = e k , where e k is the column vector of zeroes with
a 1 in the kth entry. I.e., the n×n identity matrix can be viewed as a bunch
of column vectors I n = (e 1 e 2 · · · e n ). So, putting the e k ’s together into an
identity matrix, we get:

M I ∼ I M −1 I = I M −1

  −1
−1 2 −3
Example 93 Find  2 1 0  .
4 −2 5
We start by writing the augmented matrix, then apply row reduction to the left side.

   
−1 2 −3 1 0 0 1 −2 3 1 0 0
2 1 0 0 1 0 0 5 −6 2 1 0
   
  ∼  
4 −2 5 0 0 1 0 6 −7 4 0 1
 3 1 2 
1 0 − 0
5 4 5
 0 1 − 6 2 1 0 
5 5 5 
∼ 
0 0 1 4 − 6 1
5 5 5
1 0 0 −5 4 −3
 
0 1 0 10 −7 6
 
∼  
0 0 1 8 −6 5

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