Here we're given a matrix A,
which is a two by two matrix,
and we're asked to find A squared,
where A squared, we know,
is equal to A times A.
When squaring a matrix, there are no shortcuts.
We first need to make sure
that the product is possible,
and then if it is,
we multiply like we normally do.
To see if the product is possible,
or see if it's possible to square matrix A,
because A squared is equal to A times A,
we would be multiplying a two by two matrix
by another two by two matrix.
So if the squaring is possible,
the number of columns in the first matrix
must equal the number of rows in the second matrix,
which in this case,
notice how this is true
because two equals two.
And the remaining values
are the number of rows in the first matrix,
and the number of columns in the second matrix,
tell us the dimensions of the product
is going to be a two by two matrix.
As an example of a matrix
that we can't square,
let's say we had a matrix B
that was a two by three matrix.
If we try to square matrix B,
we would have a two by three
times a two by three.
The number of columns in the first matrix
does not equal the number of rows in the second matrix,
so we cannot square a two by three matrix.
From this, we can conclude
we can only square square matrices.
So again, there are no shortcuts.
We'll go ahead and set this up
as matrix A times matrix A,
which again, we now know is going to be
another two by two matrix.
And now to find the elements in this product,
we will always multiply
rows from the first matrix
and columns from the second matrix.
So let's begin by determining the element in row one,
column one, or A sub one comma one.
To do this, we'll multiply row one
from the first matrix
and column one from the second matrix.
So here's row one from the first matrix,
and here's column one from the second matrix.
To multiply rows and columns,
we find the sum of the products
of the first elements,
second elements, third element, and so on,
but in this case,
because the rows and columns contain two elements,
we'll be summing two products.
A sub one comma one is equal
to negative three times negative three
plus negative five times two.
So we'd have positive nine
plus negative 10, or minus 10,
which equals negative one.
Next we'll find the element in row one column two,
or A sub one comma two.
So now we'll multiply row one from the first matrix
and column two from the second matrix.
So here's row one from the first matrix,
and here's column two from the second matrix.
So we'd have negative three times negative five
plus negative five times negative one,
which is equal to positive 15.
This would be plus five,
which equals 20.
Now let's find the element in row two, column one,
or A sub two comma one.
So now we multiply row two from the first matrix
and column one from the second matrix.
So again, row two from the first matrix
and column one from the second matrix.
So we have two times negative three
plus negative one times two,
which is equal to negative six
plus negative two, or minus two,
which equals negative eight.
And finally, we have A sub two comma two,
which is the element in row two, column two,
so we multiply row two from the first matrix
and column two from the second matrix.
So here's row two from the first matrix,
and here's column two from the second matrix.
So we have two times negative five
plus negative one times negative one,
which equals negative 10 plus one,
which equals negative nine.
So here's the square of matrix A.
I hope you found this helpful.