### Video Transcript

In this video, we will learn how to
combine the operations of addition, subtraction, scalar multiplication, and
transposing matrices. We will begin by recalling how we
perform each of these operations.

If 𝐴 and 𝐵 are matrices of the
same order or dimension, then they can be added or subtracted. If we let 𝐴 and 𝐵 be the
two-by-two matrices as shown, we can simply add or subtract the matrices by adding
or subtracting their corresponding components. The matrix 𝐴 plus 𝐵 will have the
same order, two by two, and have the following four components. We can calculate the matrix 𝐴
minus 𝐵 in a similar way by subtracting the corresponding components. It is important to note that matrix
𝐴 plus matrix 𝐵 is equal to matrix 𝐵 plus matrix 𝐴. However, matrix 𝐴 minus matrix 𝐵
is not necessarily equal to matrix 𝐵 minus matrix 𝐴. This means that addition of
matrices is commutative, whereas subtraction is not. The matrix 𝐴 minus 𝐵 is actually
equal to negative one multiplied by matrix 𝐵 minus 𝐴.

Let’s now recall how we can
multiply a matrix by a scalar. To multiply any matrix by a scalar,
we multiply each element or component by the scalar. We can multiply a matrix of any
order or size by a scalar. Let’s consider the two-by-two
matrix as shown. We can multiply this matrix by the
scalar 𝑘. This gives us the two-by-two matrix
as shown. Let’s now recall what we mean by
the transpose of a matrix. The transpose of a matrix is an
operator which switches the rows and columns. When dealing with a square matrix,
we simply flip the matrix over its leading or main diagonal. The transpose of a matrix is
denoted with an exponent 𝑇. This is usually written as a
capital but can sometimes be written as a lowercase 𝑡. When finding the transpose of a
two-by-two matrix, the elements in the top-right and bottom-left corners swap.

Let’s now consider the three-by-two
rectangular matrix 𝐵 as shown. It has elements 𝑢, 𝑣, 𝑤, 𝑥, 𝑦,
𝑧. As we’re switching the rows and
columns, 𝐵 transpose is a two-by-three matrix. It has two rows and three columns
with elements 𝑢, 𝑤, 𝑦, 𝑣, 𝑥, 𝑧. The first row of matrix 𝐵 is the
first column of 𝐵 transpose. Likewise, the first column of
matrix 𝐵 is the first row of 𝐵 transpose. This can be summarized as
follows. A matrix with 𝑚 rows and 𝑛
columns will have a transpose with 𝑛 rows and 𝑚 columns. In this case, the original matrix
will have order 𝑚 by 𝑛, whereas the matrix transpose will have order 𝑛 by 𝑚. This is different to perform in the
operations of addition, subtraction, and scalar multiplication to rectangular
matrices where the order remains the same.

We will now look at some examples
where we need to perform a combination of these operations.

Given that matrix 𝐴 equals
negative seven, five, negative four, negative two; matrix 𝐵 equals one, zero,
seven, negative two, what is one-third multiplied by 𝐴 plus 𝐵?

We recall that we can add two
matrices if they have the same order. In this case, both matrix 𝐴 and
matrix 𝐵 are two-by-two square matrices. In order to add two matrices of the
same order, we simply add their corresponding components or elements. Starting in the top-left corner,
negative seven plus one is equal to negative six. Five plus zero is equal to
five. Negative four plus seven is equal
to three. And finally, negative two plus
negative two is equal to negative four. The matrix 𝐴 plus 𝐵 is equal to
negative six, five, three, negative four.

We are asked to calculate one-third
of this, so we need to multiply the matrix by the scalar one-third. This can be done by multiplying
each of the elements by one-third or dividing them by three. One-third multiplied by negative
six is negative two. Repeating this process for the
other elements gives us five-thirds, one, and negative four-thirds. If the matrices 𝐴 and 𝐵 are equal
to negative seven, five, negative four, negative two and one, zero, seven, negative
two, respectively, then one-third of 𝐴 plus 𝐵 is equal to negative two,
five-thirds, one, negative four-thirds. Whilst we won’t prove it in this
video, it is important to note that one-third multiplied by the matrix 𝐴 plus 𝐵 is
equal to one-third of matrix 𝐴 plus one-third of matrix 𝐵.

In our next example, we will need
to calculate the transpose of a two-by-two matrix.

The matrix six, negative four,
negative three, two minus the transpose of the matrix five, negative three, negative
four, one is equal to what. Is it (A) one, negative eight,
negative six, one; (B) 11, zero, zero, three; (C) 𝐼; or (D) 𝑂?

Before starting this question, it
is worth recalling what options (C) and (D) represent. 𝐼 is the identity matrix. This is a square matrix with ones
on its leading or main diagonal and zeros elsewhere. The two-by-two identity matrix has
elements one, zero, zero, one. Option (D) represents the zero
matrix. All elements within this must equal
zero. Therefore, the two-by-two zero
matrix is equal to zero, zero, zero, zero.

Our question here also contains the
operation transpose. To calculate the transpose of a
matrix, we switch the rows and the columns. When dealing with a two-by-two
square matrix, we flip the matrix over its leading or main diagonal. In this example, the numbers
negative three and negative four will swap places. The transpose of the matrix five,
negative three, negative four, one is five, negative four, negative three, one. We need to subtract this from the
matrix six, negative four, negative three, two.

We recall that when subtracting two
matrices, they must be of the same order, and we simply subtract the corresponding
elements or components. Six minus five is equal to one. Negative four minus negative four
is the same as negative four plus four, which equals zero. Likewise, negative three minus
negative three is equal to zero. And two minus one is equal to
one. The matrix six, negative four,
negative three, two minus the transpose of the matrix five, negative three, negative
four, one is equal to one, zero, zero, one. As previously mentioned, this is
the identity matrix. Therefore, the correct answer is
option (C).

In our next example, we need to
find the missing matrix in the equation.

If negative three 𝑋 plus the
matrix negative three, zero, nine, 12 is equal to the zero matrix, then 𝑋 equals
what. Is it (A) one, zero, negative
three, negative four; (B) negative one, zero, three, four; (C) three, zero, negative
nine, negative 12; or (D) negative three, zero, nine, 12?

We recall that the zero matrix has
all elements equal to zero. Therefore, the two-by-two zero
matrix is equal to zero, zero, zero, zero. There are lots of ways of solving
this problem. We could divide through by three or
negative three. However, in this example, we will
let the matrix 𝑋 be the two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑. In order to multiply any matrix by
a scalar, we simply multiply each of the elements or components by that scalar. This means that the matrix negative
three 𝑋 has elements negative three 𝑎, negative three 𝑏, negative three 𝑐,
negative three 𝑑. Adding the matrix negative three,
zero, nine, 12 to this will give us the zero matrix.

We can then set up four linear
equations by comparing the corresponding components or elements. Firstly, negative three 𝑎 plus
negative three is equal to zero. We can add three to both sides such
that negative three 𝑎 is equal to three. Dividing through by negative three
gives us 𝑎 is equal to negative one. Next, we can look at the top-right
elements. This gives us negative three 𝑏
plus zero is equal to zero. As negative three 𝑏 is equal to
zero, dividing both sides by negative three gives us 𝑏 is equal to zero. We can repeat this process for the
bottom row, giving us values of 𝑐 and 𝑑 equal to three and four. The matrix 𝑋 is therefore equal to
negative one, zero, three, four. We can therefore conclude that the
correct answer is option (B).

We will now consider one final
example where we look at rectangular matrices.

Given that matrix 𝑋 equals
negative three, negative two, one, five, negative eight, negative eight; matrix 𝑌
equals negative one, eight, negative nine, negative nine, seven, negative two; and
matrix 𝑍 equals three, negative eight, negative seven, zero, negative eight, five,
what is the matrix three 𝑋 plus 𝑌 minus three 𝑍?

We recall that in order to multiply
any matrix by a scalar, we simply multiply each of the elements or components by
that scalar. This means that the matrix three 𝑋
is equal to negative nine, negative six, three, 15, negative 24, negative 24. In the same way, the matrix three
𝑍 is equal to nine, negative 24, negative 21, zero, negative 24, 15. We need to add matrix 𝑌 to three
𝑋 and then subtract three 𝑍. We can only add and subtract
matrices when they are of the same order. In this question, all three
matrices have three rows and two columns. Therefore, their order is three by
two.

We now need to add and subtract the
corresponding elements or components. Negative nine plus negative one is
negative 10. And subtracting nine from this
gives us negative 19. In the top right, we have negative
six plus eight, which is equal to two. And then subtracting negative 24
from this gives us 26. Repeating this method for our other
four elements gives us 15, six, seven, and negative 41. If 𝑋, 𝑌, and 𝑍 are the matrices
given, then the matrix three 𝑋 plus 𝑌 minus three 𝑍 is equal to negative 19, 26,
15, six, seven, negative 41. Note that this matrix has the same
order as our original matrices.

We will now summarize the key
points from this video. Matrix addition and subtraction is
only well defined between matrices of the same order. This means that we can only add or
subtract matrices if they are of the same size. Matrix addition and subtraction is
completed entry by entry. This means that we add or subtract
the corresponding components or elements. Multiplying a matrix by a scalar is
also completed entry by entry. We can multiply a matrix of any
order by any scalar. For any matrix 𝐴 of order 𝑚 by
𝑛, the matrix transpose is of order 𝑛 by 𝑚. This means that we can perform the
transpose of any matrix whether it is square or rectangular. The matrix transpose simply
switches the rows and columns. When dealing with a square matrix,
we can flip the matrix over its leading or main diagonal.