Fundamentals of Matrix Algebra, 2011a

2.5 Solving Matrix Equaons AX = B
[ ]
0 0 2 1 4
21. A =
,
−2 −1 −4 −1 5
[ ] 3 ⃗b =
4
⎡
3 0 −2 −4
⎤
5
22. A = ⎣ 2 3 2 0 2 ⎦,
−5 0 4 0 5
⎡ ⎤
−1
⃗b = ⎣ −5 ⎦
4
⎡
−1 3 1 −3 4
⎤
23. A = ⎣ 3 −3 −1 1 −4 ⎦,
−2 3 −2 −3 1
⎡ ⎤
1
⃗b = ⎣ 1 ⎦
−5
⎡
−4 −2 −1 4 0
⎤
24. A = ⎣ 5 −4 3 −1 1 ⎦,
4 −5 3 1 −4
⎡ ⎤
3
⃗b = ⎣ 2 ⎦
1
In Exercises 25 – 28, a matrix A and vector ⃗b
are given. Solve the equaon A⃗x = ⃗b, write
the soluon in vector format, and sketch the
soluon as the appropriate line on the Cartesian
plane.
[ ] [ ]
2 4 0
25. A =
, ⃗b =
−1 −2 0
[ 2 4
26. A =
−1 −2
]
, ⃗b =
[ ] −6
3
[ ] [ ]
2 −5 1
27. A =
, ⃗b =
−4 −10 2
[ ] [ ]
2 −5 0
28. A =
, ⃗b =
−4 −10 0
2.5 Solving Matrix Equaons AX = B
AS YOU READ ...
. . .
1. T/F: To solve the matrix equaon AX = B, put the matrix [ A X ] into reduced
row echelon form and interpret the result properly.
2. T/F: The first column of a matrix product AB is A mes the first column of B.
3. Give two reasons why one might solve for the columns of X in the equaon AX=B
separately.
We began last secon talking about solving numerical equaons like ax = b for x.
We menoned that solving matrix equaons of the form AX = B is of interest, but we
first learned how to solve the related, but simpler, equaons A⃗x = ⃗b. In this secon
we will learn how to solve the general matrix equaon AX = B for X.
We will start by considering the best case scenario when solving A⃗x = ⃗b; that is,
when A is square and we have exactly one soluon. For instance, suppose we want to
solve A⃗x = ⃗b where
A =
[ ] 1 1
2 1
and ⃗b =
[ 0
1
]
.
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