### Video Transcript

Given that π΄ is the matrix with
first row three, three, three and second row three, three, three and π΅ is the
matrix with first row three, three and second row three, three, is it true that the
matrix π΄ is equal to the matrix π΅?

Letβs start by recalling what we
mean when we say that two matrices are equal. If we have the entry in row π and
column π of matrix π΄ is π ππ and the entry in row π and column π of matrix π΅
is π ππ, then if π ππ is equal to π ππ for all of our values of π and π,
we say that the matrix π΄ is equal to the matrix π΅. Otherwise, we say that these
matrices are not equal. So to check the two matrices are
equal, we need to check that all of their entries are identical.

Letβs start with matrix π΄. We can see this has two rows and
three columns. So in our case, what would our
values of π ππ be? First, our matrix π΄ has two rows
and three columns, so our values of π range from one to two and our values of π
range from one to three. We can then do something similar
for our matrix π΅. If π ππ is the entry in matrix
π΅ in row π and column π, then because our matrix π΅ only has two rows and two
columns, our values of π range from one to two and our values of π also range from
one to two. But now we can start to see our
problem from our definition. The entries must be equal for all
possible rows and columns. Our matrix π΄ has three
columns. However, our matrix π΅ only has two
columns. So these matrices canβt possibly be
equal.

For example, if we highlight the
two entries in matrix π΄ in column three, by our definition of equality, we would
have to have a third column in matrix π΅ which is equal to this column. So in this case, the matrix π΄ is
not equal to the matrix π΅.

In fact, we can use exactly the
same line of reasoning as we did in this question to deduce that if two matrices
have different orders, they canβt be equal. In other words, if they have a
different number of rows or columns, they canβt be equal. This means what weβve shown is for
two matrices to be equal, they must have the same number of rows or columns. In other words, they must have the
same order.