For two matrices to be
equal, they must be of the same size and have all the same entries in
the same places. For instance, suppose you have the following two matrices:

These matrices cannot be
the same, since they are not the same size. Even if A
and B
are the following two matrices:

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...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 for matrices. 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 if A
and B
have the same number of entries or even the same numbers as entries. Unless
A
and B
are the same size and the same shape and have the same values in exactly
the same places, they are not equal.

This property of matrix
equality can be turned into homework questions. You will be given two
matrices, and you will be told that they are equal. You will need to use
this equality to solve for the values of variables.

Given that
the following matrices are equal, find the values of x
and y.

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. Then a_{1,1}
must equal b_{1,1},
a_{1,2}
must equal b_{1,2},
and so forth. The entries a_{1,2}
and a_{2,1}
are clearly equal, respectively, to entries b_{1,2}
and b_{2,1}
"by inspection" (that is, "just by looking at them").
But a_{1,1}
= 1 is not obviously
equal to b_{1,1}
= x. For A
to equal B,
I must have
a_{1,1} = b_{1,1},
so it must be that 1
= x. Similarly,
I must have a_{2,2}
= b_{2,2},
so then 4
must equal y.
Then the solution is:

Given that
the following matrices are equal, find the values of x,
y,
and z.

To have A
= B, I must
have all entries equal. That is I must have a_{1,1}
= b_{1,1}, a_{1,2} = b_{1,2},
a_{2,1} = b_{2,1},
and so forth. In particular, I must have:

4
= x –2
= y + 4
3 = ^{z}/_{3}

...as you can see from
the highlighted matrices:

Solving these three equations,
I get:

x
= 4, y = –6, and z
= 9.

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, but that's another
lesson for another
time).