Welcome to a lesson on diagonal matrices.
The goals of this video are to define a diagonal matrix,
determine if a diagonal matrix is singular or invertible,
determine the inverse of an invertible diagonal matrix,
and then finally, determine a diagonal matrix
raised to a power.
A square matrix is called a diagonal matrix
if all of the nondiagonal entries are zero,
and the main diagonal, or this diagonal here,
can be constants or zeros.
So a diagonal matrix must fit this form here.
If any of the diagonals of a diagonal matrix are zero,
or any of these elements here, the matrix is singular,
meaning it is not invertible
or does not have an inverse matrix.
If the diagonals of a diagonal matrix are nonzero,
then the inverse of the diagonal can be easily found
by using this formula here.
To find the inverse of matrix D from the previous screen,
we just need to find the reciprocals
of the main diagonal as we see here.
All other elements will still be zero.
So let's look at a couple of examples.
We're asked to determine
if the diagonal matrix has an inverse,
and then if it does, find the inverse.
We're looking at matrix A.
Notice how the main diagonal
does contain an element that's zero,
this element here, and therefore, matrix A is singular,
meaning it does not have an inverse.
But notice how the diagonal of matrix B
does not contain zeros, and therefore,
we can easily find the inverse of matrix B
by finding the reciprocal of these values here
in the main diagonal.
So the inverse of matrix B
will have a main diagonal where the element
in row one, column one will be the reciprocal of three,
which is 1/3.
The element in row two, column two will be the reciprocal
of negative four, which will be negative -1/4.
Reciprocal of one is still one,
and the reciprocal of eight would be 1/8.
All other elements will still be zero.
Remember this means that
B times B inverse is equal to B inverse times B,
which will give us a four by four identity matrix.
So, of course, if we wanted to, we could check this.
The last topic for this video will be raising
a diagonal matrix to a power.
Multiplying diagonal matrices
just requires multiplying corresponding diagonal elements.
Therefore, if we raise matrix D to the power of m,
we just need to raise each element in the main diagonal
to the power of m, as we see here.
So for example, if we want to find A to the third
given matrix A, we just need
to raise each element in the diagonal to the power of three.
So A to the third is going to be equal to the matrix
where the main diagonal is going to be negative two
to the third, four to the third,
and negative three to the third.
Of course, all other elements will remain zero.
So negative two raised to the third power
would be negative eight.
Four to the third is 64,
and negative three to the third is negative 27.
Okay, that's going to do it for this video.
I hope you found this helpful.