### Video Transcript

If the matrix π΄ is equal to one,
zero, zero, one, which of the following is true? The matrix π΄ is a row matrix. The matrix π΄ is a column
matrix. The matrix π΄ is an identity
matrix. Or the matrix π΄ is a zero
matrix.

Letβs consider each of these four
statements and remind ourselves of the features of the type of matrix they refer
to. Weβll start with the final
statement: the matrix π΄ is a zero matrix. A zero matrix can have any
dimensions, but every element within the matrix must be equal to zero. Looking at our matrix π΄, we can
see that it does indeed have two elements which are equal to zero. But the other two elements, the
elements which are on what we call the leading diagonal, are not equal to zero. So matrix π΄ is not a zero matrix
because it doesnβt have every element equal to zero.

Letβs now consider the top
statement: the matrix π΄ is a row matrix. A row matrix or row vector is a
matrix of order one by π. Itβs simply a matrix which has only
one row, although it can have any number of columns. In its general form, it would look
something like this, and the elements would be labeled as π one one, π one two,
all the way up to π one π. Looking at our matrix π΄, we can
see that it has not one but two rows. And therefore, the matrix π΄ is not
a row matrix.

Weβll now consider the second
statement: the matrix π΄ is a column matrix. Well, a column matrix or a column
vector is a matrix of order π by one. So it can have any number of rows,
but it must have only one column. In its general form, it would look
a little something like this. And we could label the elements as
π one one, π two one, all the way down to π π one. Looking at our matrix π΄, itβs easy
to see that this matrix has two columns, not one. And therefore, the matrix π΄ is not
a column matrix.

Weβre left with only one
possibility. Is the matrix π΄ an identity
matrix? Well, an identity matrix is, first
of all, a diagonal matrix which also means it must be square. Being a diagonal matrix means that
all of the elements not on the leading diagonal, thatβs the diagonal from the top
left to the bottom right, must be equal to zero. We can see that this is the case
for our matrix π΄. Furthermore, in order to be an
identity matrix, rather than just a diagonal matrix, all of the elements on the
leading diagonal must be equal to one. Looking at our matrix π΄, we can
see that this is true for the two elements on the leading diagonal.

So we have a diagonal matrix. All of the elements not on the
leading diagonal are equal to zero. And all of the elements that are on
the leading diagonal are equal to one, which means that the matrix π΄ is an identity
matrix.

In fact, because itβs a matrix of
order two by two, we sometimes refer to it as πΌ two, the two-by-two identity
matrix. The only statement which is true
then is this one: the matrix π΄ is an identity matrix.