MATH 1324
LAB 6
DUE:
Inverse of a Matrix and Applications
You can find the multiplicative inverse of a matrix using Excel.
⎡ 1
EXAMPLE: Let A = ⎢ −1
⎢
⎢⎣ −1
2
5
3
3⎤
6 ⎥ . Find A−1 .
⎥
3 ⎥⎦
Enter the matrix A into your spreadsheet.
Recall that in order to find the inverse of a matrix, the original matrix must be a
square matrix. In this example A is a 3 × 3 , so we can try to find the inverse of A
(keep in mind that not every square matrix has an inverse). Highlight an empty
group of cells that has three rows and three columns. Type =MINVERSE(, then
highlight the cells the make up the matrix A, type a right parenthesis ), then press
Ctrl-Shift-Enter.
Note: The formula should look something like (assuming that the entries of the
matrix A is start at B2, for the upper left corner, and finish at D4, for the lower
right hand corner.)
=MINVERSE(B2:D4)
MATH 1324
LAB 8
PAGE 2
The result is
We can check our result by multiplying A by A−1 . Recall that AA−1 = I = A−1 A .
Remember to use the MMULT command to multiply matrices.
Entries like -4.4409E-16 are in Excel's from of scientific notation. In this case, the
number 4.4409E-16 is equivalent to
−4.4409 × 10−16 = −0.00000000000000044409 ≈ 0
Thus the three entries below the diagonal of 1's are all 0. The result is a 3 × 3
identity matrix. Recall that we can have Excel round our results to a desired
number of places after the decimal using the ROUND command. The format is
ROUND(expression, number of places after decimal)
Multiply A−1 A , and round your results to 6 places after the decimal.
Highlight and empty group of cells that has three rows and three columns. Type
=ROUND(MMULT(, highlight the cells of the matrix A−1 , followed by a comma,
then highlight the cells of the matrix A, followed by a right parenthesis ). Then
type a comma, followed by a 6, and type another right parenthesis ), then press
Shift-Ctrl-Enter.
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LAB 8
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Assuming that the entries of the matrix A is start at B2 and finish at D4, and the
entries for A−1 start at B7 and end at D9, the command should be
=ROUND(MMULT(B2:D4, B7:D9), 6)
EXERCISES
1.
2.
4⎤
⎡ −1 −1
⎢
Let A = 3
3 −22 ⎥ , find A−1 . Round the entries of the matrix to six places
⎢
⎥
19 ⎥⎦
⎢⎣ −2 −1
after the decimal. Verify your result by showing that AA−1 = I = A−1 A .
Solve the system of equations by using inverses.
x+ y
a.
2x + 3y − z = 0
+ 2z = 4
x
x+ y
b.
3.
=2
= 0
2x + 3y − z = 4
+ 2 z = −2
x
A simplified economy is based on four segments: Petroleum (P), textiles (TX),
transportation (TR), and chemicals (CH). One unit of petroleum output requires
0.1 unit of its own output, 0.2 unit of transportation output, and 0.4 unit of
chemical output. One unit of textile output requires 0.1 unit of its own input, 0.4
unit of petroleum output, 0.15 unit of transportation output, and 0.3 unit of
chemical output. One unit of transportation output requires 0.1 unit of its own
output, 0.6 unit of petroleum output, and 0.25 unit of chemical output. One unit of
chemical output requires 0.2 unit of its own output, 0.2 unit of petroleum output,
0.1 unit of textile output, and 0.3 unit of transportation output.
MATH 1324
LAB 8
PAGE 4
a.
What is the input-output matrix for this problem?
b.
The current demand for the four segments is 1000 units of petroleum, 350
units of textiles, 925 units of transportation, and 900 units of chemicals.
How many units of should each segment of the economy produce to meet
the current demand?
c.
If the demand for petroleum increases by 20% and the demand for
transportation increases by 15%, now many units of should each segment
of the economy produce to meet the new demand?