### Video Transcript

Given that π΅ is the matrix of
elements one, negative three, seven, negative three and π΅ minus π΅ transpose all
transpose is equal to π΄, determine the value of π one two plus π two one.

Letβs begin by clarifying some of
the notation in the question. These superscript πs are used to
denote the transpose of a matrix. The transpose of a matrix is the
matrix found by swapping the rows and columns of that matrix around. For example, the first column of a
matrix π΅ becomes the first row of the matrix π΅ transpose, and so on. Weβll look at how to find the
transpose of these particular matrices during the question.

The notation lowercase π, and then
a subscript one and two means the element in the first row and second column of the
matrix capital π΄. And in the same way, the notation
lowercase π and then a subscript two one means the element in the second row and
first column of the matrix capital π΄.

Weβre told that the matrix π΄ is
equal to π΅ minus π΅ transpose all transpose. So, in order to determine the value
of this sum, we need to find the matrix π΄. Letβs begin by finding the matrix
π΅ transpose. So, this is the transpose of the
matrix with elements one, negative three, seven, negative three.

The transpose is found by swapping
the rows and columns around. So, the first row of the matrix π΅,
thatβs the row one, negative three becomes the first column of the matrix π΅
transpose. And then the second row of the
matrix π΅ becomes the second column of the matrix π΅ transpose. So, π΅ transpose is the matrix of
elements one, seven, negative three, negative three.

Next, we need to find the matrix π΅
minus π΅ transpose. So, we take the matrix weβve just
found for π΅ transpose and subtract it from our matrix π΅. To subtract two matrices, which
must be of the same order, we subtract corresponding elements. So, in the first row and first
column we have one minus one. In the first row and second column,
we have negative three minus seven. In the second row first column, we
have seven minus negative three. And in the second row second
column, we have negative three minus negative three.

Evaluating each of these
subtractions gives the matrix with elements zero, negative 10, 10, zero. Then the matrix π΄ is the transpose
of this matrix, so the transpose of the matrix zero, negative 10, 10, zero.

The first row becomes the first
column of the transpose matrix, giving zero, and then negative 10. And then the second row becomes the
second column of the transpose matrix. So, the matrix π΅ minus π΅
transpose all transpose is equal to zero, 10, negative 10, zero.

Remember we are asked to determine
the value of π one two plus π two one. The element in the first row and
second column is equal to 10. And the element in the second row
first column is equal to negative 10. So, the sum of these two elements,
10 plus negative 10, is equal to zero. Weβve found then that if π΅ is the
matrix one, negative three, seven, negative three and π΄ is the matrix π΅ minus π΅
transpose all transpose, then the value of π one two plus π two one is equal to
zero.