Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
by the number 0, the result is the zero matrix, or 0.
We began this secon with the concept of matrix equality. Let’s put our matrix
addion properes to use and solve a matrix equaon.
. Example 23 Let
Find the matrix X such that
A =
[ ] 2 −1
.
3 6
2A + 3X = −4A.
S We can use basic algebra techniques to manipulate this equaon
for X; first, let’s subtract 2A from both sides. This gives us
Now divide both sides by 3 to get
3X = −6A.
X = −2A.
Now we just need to compute −2A; we find that
[ ] −4 2
X =
.
−6 −12
.
Our matrix properes idenfied 0 as the Addive Identy; i.e., if you add 0 to any
matrix A, you simply get A. This is similar in noon to the fact that for all numbers a,
a + 0 = a. A Mulplicave Identy would be a matrix I where I × A = A for all matrices
A. (What would such a matrix look like? A matrix of all 1s, perhaps?) However, in
order for this to make sense, we’ll need to learn to mulply matrices together, which
we’ll do in the next secon.
Exercises 2.1
Matrices A and B are given below. In Exercises
1 – 6, simplify the given expression.
A =
1. A + B
2. 2A − 3B
3. 3A − A
4. 4B − 2A
[ ] 1 −1
7 4
B =
[ ] −3 2
5 9
5. 3(A − B) + B
6. 2(A − B) − (A − 3B)
Matrices A and B are given below. In Exercises
7 – 10, simplify the given expression.
[ ] [ ]
3 −2
A = B =
5 4
7. 4B − 2A
8. −2A + 3A
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