### Video Transcript

In this lesson, we will learn how
to use some properties of the matrix inverse. At this point, you should be
familiar with finding the determinant and the inverse of two-by-two and
three-by-three matrices. Firstly, before we get started,
recall the identity matrix, the matrix whose elements along the principal diagonal,
that is, the diagonal from top left to bottom right, are one and the rest are
zero. So the two-by-two identity matrix
is one, zero, zero, one. And the three-by-three identity
matrix is one, zero, zero, zero, one, zero, zero, zero, one.

Recall that a square matrix π΄ is
called invertible if there is an π΄ inverse such that π΄ multiplied by π΄ inverse is
the same as π΄ inverse multiplied by π΄, which gives us the identity matrix. So what we can say from this
definition is that if π΅ is the inverse of π΄, then the product π΄π΅ and the product
π΅π΄ gives us the identity matrix. And we can use this to check
whether two matrices π΄ and π΅ are the inverse of each other.

So letβs now see the properties of
the matrix inverse. From the definition of the matrix
inverse, we have that the inverse of the inverse of π΄ is just π΄. Secondly, we have that the inverse
of the product π΄π΅ is the inverse of π΅ multiplied by the inverse of π΄. Weβve got to be really careful with
this one because it might be easy to think that this is the inverse of π΄ multiplied
by the inverse of π΅. But we know that matrix
multiplication is not commutative.

We can do a quick proof to show how
this works. Remember, we said that if we
multiply a matrix by its inverse, we get the identity matrix. So if we take the matrix π΄π΅ and
multiply it by its inverse, which is π΅ inverse multiplied by π΄ inverse, we should
get the identity matrix. So letβs double-check this. Because of the associative property
of matrix multiplication, this is the same as π΄ multiplied by π΅ multiplied by π΅
inverse multiplied by π΄ inverse. And we know that π΅ multiplied by
π΅ inverse must give us the identity matrix, because thatβs just from the definition
of an invertible matrix. So we have π΄ multiplied by the
identity matrix multiplied by π΄ inverse. But multiplying any matrix by the
identity matrix just gives us the same matrix. So this is just π΄ multiplied by π΄
inverse. And again from the definition of an
invertible matrix, we know that π΄ multiplied by π΄ inverse just gives us the
identity matrix.

So because when we multiply π΄π΅ by
its inverse, which is π΅ inverse multiplied by π΄ inverse, and we get the identity
matrix, weβve shown that π΅ inverse multiplied by π΄ inverse is definitely the
inverse of π΄π΅. And for a conclusive proof, we
could show by the same method that π΅ inverse multiplied by π΄ inverse multiplied by
π΄π΅ also gives us the identity matrix.

Letβs move on and look at the third
property of the matrix inverse. π΄ transposed inverse is equal to
π΄ inverse transposed. Remember, this notation π means
the matrix transpose. And we transpose a matrix by
swapping the rows with the columns. As an example, if we have the
matrix π΄ one, four, six, two, then the matrix π΄ transposed is equal to one, six,
four, two. Again, we can check this result
because we know that we can take a matrix and multiply it by its inverse to get the
identity matrix. So if we take the matrix π΄
transposed and multiply it by its inverse, which is π΄ inverse transposed, we should
get the identity matrix.

To proceed from here, we recall the
property of the matrix transpose. The transpose of π΄π΅ is π΅
transpose multiplied by π΄ transpose. That means this is equal to π΄
inverse multiplied by π΄ transposed. But from the definition of the
matrix inverse, we know that π΄ inverse multiplied by π΄ gives us the identity
matrix. So we get the identity matrix
transpose. But if we transpose the identity
matrix, we just get the identity matrix. So weβve shown that multiplying π΄
transpose by π΄ inverse transpose gives us the identity matrix, which confirms this
property. And for a conclusive proof by a
similar method, we can show that π΄ inverse transpose multiplied by π΄ transpose
also gives us the identity matrix.

And finally we have that π΄ to the
πth power inverse is equal to π΄ inverse to the πth power. Again, we can show this by proving
that π΄ to the πth power multiplied by π΄ inverse to the πth power is going to
give us the identity matrix. We can see that weβre going to be
able to pair off the π΄βs with the π΄ inverse. And each time we do this, weβll get
the identity matrix. We know that we have the same
number of π΄βs as π΄ inverses. So by pairing these off and
multiplying them together, we know weβre going to end up with the identity
matrix. And by a similar method, we can
show that the inverse of π΄ to the πth power multiplied by π΄ to the πth power
also gives us the identity matrix. So letβs now see how we can use
these properties to answer some questions.

If π΄ is a matrix, which of the
following is equal to π΄ inverse squared? π΄ to the half power, π΄ squared,
π΄ inverse to the half power, or π΄ squared inverse.

We can answer this question by
recalling the property of inverse matrices. That is, π΄ to the πth power
inverse equals π΄ inverse to the πth power, for π is a positive integer. So with this in mind, we can say
that π΄ inverse squared is equal to π΄ squared inverse. But letβs double-check this
relation just to be sure. We found that the inverse of π΄
squared is π΄ inverse squared. And we know if we take a matrix and
multiply it by its inverse, we should get the identity matrix. So letβs check this.

If we take the matrix π΄ squared
and multiply it by its inverse π΄ inverse squared, we should get the identity
matrix. We can just write this as π΄
multiplied by π΄ multiplied by π΄ inverse multiplied by π΄ inverse. And because of the associativity
property of matrix multiplication, we can write it in this way. We know π΄ multiplied by π΄ inverse
gives us the identity matrix. And we know multiplying any matrix
by the identity matrix just gives us the same matrix. So this is just π΄ multiplied by π΄
inverse, which gives us the identity matrix.

If π΄ is a matrix, which of the
following is equal to π΄ inverse transpose?

Recall that this notation means the
transpose of a matrix. This just means we switch the rows
with the columns. For example, if we have the matrix
π equals two, one, six, seven, π transpose is equal to two, six, one, seven. To answer this question, we recall
the property of the matrix inverse. That is, π΄ transposed inverse is
equal to π΄ inverse transposed. And this gives us that our answer
is the first option. π΄ inverse transposed is equal to
π΄ transposed inverse. So what weβre saying is that if we
take the matrix π΄, find its inverse, and then transpose it, this is exactly the
same result we would get by taking the matrix π΄, transposing it, and then finding
the inverse.

Consider the matrix π΄ equals
negative three, one, negative two, five. Find π΄ inverse inverse.

Recall the definition of the
inverse of π΄ is the matrix such that π΄ multiplied by π΄ inverse equals the
identity matrix. We do have a method for finding the
inverse of a two-by-two matrix. That is, given the matrix π equals
π, π, π, π, π inverse is equal to one over ππ minus ππ multiplied by the
matrix π, negative π, negative π, π. So to find the inverse of the
inverse of π΄, we could use this method to find the inverse of π΄ and then repeat
the method to find the inverse of that.

However, we do have one property of
the matrix inverse that can help us to do this a little bit quicker. That is, the inverse of the inverse
of π΄ is just equal to π΄. So if we take a matrix and find its
inverse and then invert that matrix, we get the original matrix back. So, actually, the inverse of the
inverse of our matrix π΄ is just the matrix π΄ negative three, one, negative two,
five.

Given the matrices π΄ and π΅, where
π΄ equals one, negative two, three, zero, negative one, four, zero, zero, one and π΅
equals one, negative two, five, zero, negative one, four, zero, zero, one, find
π΄π΅. And the second part of the question
says, βWithout doing any further calculations, find π΄ inverse.β

So the first thing weβre going to
do here is find the product π΄π΅. Using the usual method for
multiplying three-by-three matrices together, we find that π΄π΅ is one, zero, zero,
zero, one, zero, zero, zero, one. And we notice that this is actually
the three-by-three identity matrix. So what does this mean for our
matrices π΄ and π΅?

Well, the definition of the inverse
matrix is that itβs the π΄ inverse such that π΄ multiplied by π΄ inverse equals the
identity matrix. So the fact that we found the
product π΄π΅ to be the identity matrix means that the matrix π΅ must be the inverse
of the matrix π΄.

The second part of the question
says, βWithout doing any further calculations, find π΄ inverse.β Well, because when we find the
product π΄π΅ we get the identity matrix, this means that the matrix π΅ is the
inverse of π΄. Therefore, π΄ inverse is the matrix
π΅, which is one, negative two, five, zero, negative one, four, zero, zero, one.

Given that π΄π΅ inverse equals
one-sixth multiplied by five, negative three, negative 33, 21 and π΄ equals negative
two, negative one, negative three, negative two, determine π΅ inverse.

Letβs start with a quick reminder
of what the matrix inverse is. The inverse of a square matrix π΄,
π΄ inverse, is the matrix such that π΄ multiplied by π΄ inverse gives us the
identity matrix. And one property of the matrix
inverse which is going to prove useful to us here is π΄π΅ inverse is equal to π΅
inverse multiplied by π΄ inverse. So because weβre told the inverse
of the product π΄π΅ is one-sixth multiplied by the matrix five, negative three,
negative 33, 21, then from this property of the matrix inverse, we can say that this
is the same as π΅ inverse multiplied by π΄ inverse. But how is this going to help us
find π΅ inverse?

Well, thereβs a little bit of a
trick that we can apply here. And all it requires is remembering
the definition of the matrix inverse. We can find the matrix π΅ inverse,
π΄ inverse, π΄ by multiplying π΅ inverse, π΄ inverse by π΄ on the right. So letβs now multiply these two
matrices together to see what we get. We do this in the usual way of
multiplying two two-by-two matrices together. And then we can simplify each
entry. And we end up with one-sixth
multiplied by the matrix negative one, one, three, negative nine. We then remember when we have a
scalar multiplied by a matrix, we can just multiply each entry by that scalar. And that gives us the matrix
negative one over six, one over six, one over two, negative three over two. But how has this actually helped us
find the matrix π΅ inverse?

Well, what we found is the matrix
π΅ inverse multiplied by π΄ inverse multiplied by π΄. And from the definition of the
matrix inverse, π΄ inverse multiplied by π΄ gives us the identity matrix. So what weβve actually found is the
matrix π΅ inverse multiplied by the identity matrix. But multiplying any matrix by the
identity matrix just gives us that matrix. So what weβve found is the matrix
π΅ inverse. So by using the definition of the
matrix inverse and one of the properties of the matrix inverse, we were able to find
an unknown using the matrix inverse.

Letβs now summarize the key points
from this lesson. For square matrices π΄ and π΅, we
have that the inverse of the inverse of π΄ is just equal to π΄. The inverse of the product π΄π΅ is
equal to π΅ inverse multiplied by π΄ inverse. π΄ transpose inverse is equal to π΄
inverse transpose. So whether we transpose the matrix
and then invert it or invert the matrix and then transpose it, we get the same
result. And finally, raising the matrix π΄
to the πth power and then inverting it gives us the same result as inverting the
matrix π΄ and then raising it to the πth power. And thatβs for a positive integer
π.