Video Transcript
In this video, we’re going to
learn how to add and subtract matrices using the properties of their addition and
subtraction.
We recall that a matrix is an array
of numbers. They are arranged into rows and
columns, and each number is then referred to as an element. The dimensions of a matrix give the
number of rows and columns, so this matrix on screen has two rows and three columns
and is therefore a two-by-three matrix. With all this in mind, matrix
addition, and indeed subtraction, is fairly straightforward. To add or subtract a pair of
matrices, we simply add or subtract their corresponding elements. With this, though, we must impose a
restriction. Take this pair of two-by-two
matrices. We can see that the first element
in the sum of these matrices is 𝑎 plus 𝑒. The second element in the first row
is 𝑏 plus 𝑓 and so on.
Now it follows that if we’re adding
the elements in the matrices, we need to make sure that each element in our first
matrix has something to add to in the second and vice versa. We can therefore only add or
subtract matrices whose order is the same. They absolutely must have the same
number of rows and columns for us to be able to perform matrix addition or
subtraction. Now, matrices also have many of the
properties of addition and subtraction that real numbers do. In particular, matrix addition is
commutative; that is, it can be done in any order. It also satisfies the additive
identity property. And, that is if we add the zero
matrix, the matrix whose elements are all zero, we end up with the original
matrix. So let’s have a look at a really
simply example.
Evaluate the matrix whose elements
are eight, 11, negative three, seven plus the matrix whose elements are 10, negative
one, three, one.
We know that to add matrices, we
simply add their corresponding elements. And, of course, this only works if
the order of each matrix is the same, in other words, if each matrix has the same
number of rows and columns. Now, both of our matrices have two
rows and two columns. Their order is two by two. Now the order is the same, so we
know we can continue and add the matrices. It’s convention, of course, to
begin with the element in the first row and first column. So we’re going to begin with eight
in our first matrix and 10 in our second. And so the element in the first row
and first column of the sum of these matrices will be eight plus 10, which is of
course 18. We now move on to the element in
the first row but the second column. So that’s 11 in our first matrix
and negative one in our second. Their sum then is 11 plus negative
one. That’s the same as 11 minus one,
which is simply 10.
We’ll now move on to the elements
in the second row. We’ll begin with negative three in
our first matrix and three in our second. This time, their sum is negative
three plus three, which is zero. And so the first element in our
second row of the sum of these matrices is zero. There’s one pair of elements left
to deal with, and that’s this seven in the first matrix and one in the second. And of course their sum is seven
plus one, which is eight. The sum of our two matrices then is
the matrix whose elements are 18, 10, zero, and eight.
Now that we’ve looked at how to add
a pair of matrices, let’s look at how we subtract a pair.
Find the matrix seven, nine,
negative five, zero minus the matrix eight, negative five, two, zero.
Now, we know that we can add or
subtract matrices as long as their order is the same. We have a pair of two-by-two
matrices here, so we can continue to the next step. To subtract a pair of matrices, we
simply subtract the corresponding elements. Conventionally, we start with the
first element on the first row. So in our first matrix, that’s
seven and in our second matrix, it’s eight. The corresponding element in the
result of the difference of these two matrices then is seven minus eight, which is
negative one. Now, of course, matrix subtraction,
just like subtraction with real numbers, is not commutative. We can’t change the order, so we
absolutely do need to do seven minus eight rather than the other way round.
The element in the first row and
second column is found by working out nine minus negative five. Now, of course, when we subtract a
negative, that’s the same as adding a positive. So we’re going to work out nine
plus five, which is equal to 14. Next, we’re going to deal with the
first element in the second row. So in our first matrix, that’s
negative five, and in our second, it’s two. So we’re going to have negative
five minus two, which is negative seven. Our very final element in the
difference between our two matrices is just zero minus zero, which is, of course,
zero. And so the result of subtracting
our two matrices is a two-by-two matrix whose elements are negative one, 14,
negative seven, zero.
We’re now going to look at how to
find an unknown matrix by applying operations involving the zero matrix.
Given that 𝑋 plus the matrix whose
elements are negative six, negative eight, six, five equals zero, where zero is the
two-by-two zero matrix, find the value of 𝑋.
We begin, of course, by recalling
what we mean when we say that a matrix is a zero matrix. This is a square matrix, in other
words, a matrix with an equal number of rows and columns whose entries or elements
are all equal to zero. And so in this case, the two-by-two
zero matrix is the matrix shown. We can therefore rewrite our matrix
equation as 𝑋 plus negative six, negative eight, six, five equals zero, zero, zero,
zero. Now we’re trying to find the value
of 𝑋. So given the equation we have now
written, what can we infer about 𝑋? Well, one thing that we know is
that if we add a pair of matrices, we simply add the elements. And we can only do that if the
matrices are of the same order. We couldn’t add, for example, a
two-by-two matrix to a two-by-three matrix, nor could we add a two-by-two matrix to
just a number with a single value.
And so, for this matrix equation to
make sense, 𝑋 must also be a two-by-two matrix. Now, to solve this matrix equation,
we’re actually going to perform a similar set of steps to solving a normal
equation. We’re going to subtract this matrix
negative six, negative eight, six, five from both sides of the equation. When we subtract it from the
left-hand side, of course, we’re just going to end up with the matrix 𝑋. And so 𝑋 is zero, zero, zero, zero
minus the matrix whose elements are negative six, negative eight, six, five. Now, just like when we add a pair
of matrices and we add their elements to subtract a pair of matrices, we subtract
their individual elements.
The element in the first row and
first column then will be zero minus negative six. Now, of course, subtracting a
negative is the same as adding a positive. So that’s the same as doing zero
plus six, which is six. Then, we do zero minus negative
eight. And once again, that’s the same as
doing zero plus eight, which is eight. We now move on to the elements in
the second row, so we work out zero minus six. And, of course, that’s simply
negative six. And, finally we’re going to work
out zero minus five, and that’s negative five. And so the two-by-two matrix 𝑋 is
six, eight, negative six, negative five.
Now, this example illustrates
something really important. We see that matrix addition
satisfies the additive inverse property. Now, the additive inverse of a
number is what you add to a number to create zero, and usually that’s just found by
changing the sign of the original number. If we look at our example here, we
can see we’ve changed the sign of the individual elements. And so the matrix six, eight,
negative six, negative five is the additive inverse of the matrix negative six,
negative eight, six, five.
We’re now going to consider another
example that involves solving a matrix equation.
Consider the matrix 𝐴 which has
the elements negative one, negative one, five, 13, negative one, zero. Suppose the sum of the matrices 𝐴
and 𝐵 is 𝐴 plus 𝐵 equals one, zero, negative one, zero, one, two. Find the matrix 𝐵.
We’re given that the sum of the
matrices 𝐴 and 𝐵 is a two-by-three matrix. It has two rows and three
columns. We also know that we can add
matrices by simply adding their individual elements. But that means that the matrices
that we add must be of the same order. And so to be able to add matrix 𝐴
and 𝐵, where 𝐴 is also a two-by-three matrix and the result is a two-by-three
matrix, 𝐵 itself must be a two-by-three matrix. Now we can solve this matrix
equation as we would any other normal equation. We’re going to subtract 𝐴 from
both sides. But of course 𝐴 is a matrix, so we
can write this as 𝐵 equals one, zero, negative one, zero, one, two minus the matrix
whose elements are negative one, negative one, five, 13, negative one, and zero.
Now, just like when we add
matrices, we can subtract matrices by subtracting their individual elements. And so let’s begin with the first
element in our first matrix and the first in our second. It’s one minus negative one. But of course, when we subtract a
negative, that’s the same as adding a positive. So this is like working out one
plus one, which is equal to two. We repeat this process with the
second element in each matrix. We’re now working out zero minus
negative one, which is the same as zero plus one, which is one. We move on to the third
element. It’s negative one in the first
matrix and five in the second. So we do negative one minus five,
which is negative six.
We’ll now repeat this with the
elements on the second row. We have zero minus 13, which is
negative 13. Next, we calculate one minus
negative one. And, of course, that’s the same as
one plus one, which is two. And then our final calculation is
two minus zero, which is, of course, simply two. And so we’ve found the matrix
𝐵. It’s the two-by-three matrix with
the elements two, one, negative six, negative 13, two, two.
In our final example, we’ll
consider how we can extend this into working with more than two matrices.
Given that 𝐴 is equal to the
matrix four, three, negative one, three; 𝐵 is negative one, zero, two, three; 𝐶 is
negative five, one, zero, seven, find 𝐴 plus 𝐵 minus 𝐶.
Let’s rewrite this expression using
the matrices given. It’s four, three, negative one,
three plus negative one, zero, two, three minus negative five, one, zero, seven. Now we know how to add and subtract
matrices. As long as the matrices are of the
same order, in other words, they have the same number of rows and the same number of
columns, we add or subtract their corresponding elements. But just like working with real
numbers, addition is commutative. In other words, it can be done in
any order, but subtraction isn’t. We also know that when we consider
the order of operations, if we have addition and subtraction in the same
calculation, we simply move from left to right. And so let’s begin by taking the
first element in the first row of each matrix. That’s four, negative one, and
negative five.
We use subscript notation to define
the first element in the first row as 𝑎 sub one one. And so we get four plus negative
one minus negative five. Four plus negative one is three,
and then we’re going to subtract negative five. So it’s the same as adding
five. So we get three plus five, which is
equal to eight. So that’s the first element in the
first row. Then 𝑎 sub one two is the element
in the first row and second column. So it’s three plus zero minus
one. Three plus zero is three, and then
we do three minus one, which is equal to two. We now move on to considering the
element in the second row and the first column. It’s negative one plus two minus
zero. And since negative one plus two is
one, we get one minus zero, which is simply one.
Our final element is defined by 𝑎
sub two two. That’s the element in the second
row and second column. It’s three plus three minus
seven. Three plus three is equal to six,
so we work out six minus seven as negative one. And so all that’s left is to put
this back into its matrix form. 𝐴 plus 𝐵 minus 𝐶 is the
two-by-two matrix whose elements are eight, two, one, negative one.
We’re now going to recap the key
points from this lesson. In this video, we learned that we
can add or subtract matrices by adding or subtracting their corresponding
elements. We also saw, though, that this only
works if the order of each matrix is the same, in other words, if they have the same
number of rows and the same number of columns. We saw that matrix addition is
commutative; it can be done in any order, unlike matrix subtraction. And just like working with real
numbers, matrix addition also satisfies the additive identity property and the
additive inverse property.
