Matrix Equality
Purplemath
Where do matrices come from?
Originally, matrices were developed within the context of solving systems of linear equations. Naturally, the Babylonians were among those getting an early start on things, though the Chinese were (unbeknownst to either group) hot on their heels.
Matrices were sort of abandoned for the better part of two millenia, before being resurrected in the late 1600's.
(The well-regarded MathHistory site has more historical info, if you're interested.)
You can do many things with matrices, including adding them and multiplying them. You'll learn about those topics later. Here, we'll start with a simple question:
What does it mean for matrices to be equal?
For two matrices to be equal, they must be of the same size and have all the same entries in the same places. That is, they must have the same number of rows, the same number of columns, and the exact same numbers, in the exact same order, in their grids.
For instance, suppose you have the following two matrices:
![A = [ [ 1 3 ] [ -2 0 ] ] and B = [ [ 1 3 0 ] [-2 0 0 ] ]](matrices/defs18.gif){kind=small}
These matrices cannot be equal because they are not even the same size or shape. Even if A and B are the following two matrices:
![A = [ [ 1 4] [ 2 5 ] [ 3 6 ] ] and B = [ [ 1 2 3 ] [ 4 5 6 ] ]](matrices/defs19.gif)
...they are still not the same. Yes, A and B each have six entries, and the entries are even the same numbers, but that is not enough to fulfill the conditions of matrix equality. A is a 3×2 matrix and B is a 2×3 matrix, and, for matrices, 3×2 does not equal 2×3.
It doesn't matter that A and B have the same number of entries or even the same numbers as entries. Unless A and B are the also same size and the same shape and have the same values in exactly the same places, they can not be equal.
These properties of matrix equality can be turned into homework questions. In such a case, you will be given two matrices, and you will be told that they are equal. You will then need to use this fact of equality in order to solve for the values of variables.
- Given that the following matrices are equal, find the values of x and y.
![A = [ [ 1 2 ] [ 3 4 ] ] and B = [ [ x 2 ] [ 3 y ] ]](matrices/defs20.gif)
For A and B to be equal, they must have the same size and shape — which they do; they're each 2×2 matrices — and they must have the same values in the same spots.
This means that the a1,1 entry must equal the b1,1 entry, the a1,2 entry must equal the b1,2 entry, and so forth. So let's look at these matrices entry-wise.
The entries a1,2 and a2,1 are clearly equal, respectively, to entries b1,2 and b2,1 "by inspection" (that is, "just by looking at them").
But a1,1 = 1 is not obviously equal to b1,1 = x. For A to equal B, though, I *must* have a1,1 = b1,1. Therefore, it must be true that 1 = x.
Similarly, I must have a2,2 = b2,2, so then 4 must equal y. Then the solution is:
x = 1, y = 4
Yes, it's really as simple as comparing positions and copying numbers.
- Given that the following matrices are equal, find the values of x, y, and z.
![A = [[ 4 0 ][ 6 -2 ][ 3 1 ]] and B = [[ x 0 ][ 6 y+4 ][ z/3 1 ]]](matrices/defs21.gif)
To have A = B, I must have all entries equal. That is, I must have a1,1 = b1,1, a1,2 = b1,2, a2,1 = b2,1, and so forth. In particular, I must have:
4 = x
−2 = y + 4
...as you can see from the highlighted matrices:
Solving these three equations, I get:
x = 4
y = −6
z = 9
And that's pretty much all there is to this sort of matrix equalities.
Don't let matrices scare you. Yes, they're different from what you're used to, but they're not so bad — at least not until you try to multiply them, and that's another lesson for another time.
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