### Video Transcript

Given that the matrix π΅ equals
zero, negative two π₯, negative 63, π§ plus 14, zero, three π§, negative three π¦
plus six π₯, negative 12, zero is skew-symmetric, find the value of π₯ plus π¦ plus
π§.

Given that this matrix is
skew-symmetric, it must be true that π΅ transpose is equal to negative π΅. This holds for all skew-symmetric
matrices. And itβs a really important
property that weβre going to use to answer this question.

In order to use this, letβs first
remind ourselves what π΅ transpose means. For an π-by-π matrix, π΅ defined
by π΅ equals π΅ ππ, the transpose of π΅, denoted π΅ T, is equal to π΅ ππ. Notice how transposing a matrix
swaps the row and column indices. This means that the rows become the
columns and the columns become the rows.

So, in order to use this property,
weβre going to have to transpose the matrix π΅. We do this by firstly taking the
first row of matrix π΅, which is zero, negative two π₯, negative 63. And these entries become the first
column of the matrix π΅ transpose, like so. Then, we take the entries in the
second row of matrix π΅. Thatβs π§ plus 14, zero, three
π§. And these entries become the second
column of the matrix π΅ transpose. And finally, we take the entries in
the third row of matrix π΅. Thatβs negative three π¦ plus six
π₯, negative 12, and zero. And these entries become the
entries of the third column in matrix π΅ transpose. So that gives us the matrix π΅
transpose.

So now, weβre going to use this
property: π΅ transpose equals negative π΅. And remember, this holds true
because we know that the matrix π΅ is skew-symmetric. Well, we just found the matrix π΅
transpose, and weβre going to set this equal to negative matrix π΅. Making this matrix negative changes
the sign of each entry or term in the matrix. So the zero entries will stay the
same, and these four entries are going to change sign. Weβve got to be careful with these
two entries that have two parts with them because the negative applies to both of
the values in the parentheses.

And now we can look at equating
these two matrices. We can do this by comparing the
entries in the same positions in each matrix. For instance, because these two
matrices are equal, that must mean that π§ plus 14 is equal to two π₯. It also means that negative three
π¦ plus six π₯ is equal to 63 and that negative 12 is equal to negative three
π§. And that one is probably a good
place to start. If negative 12 is equal to negative
three π§, then π§ must be equal to four.

And then we can use that to solve
the top equation: π§ plus 14 equals two π₯. Substituting π§ equals four gives
us that four plus 14 equals two π₯, which is 18 equals two π₯. And that gives us that π₯ is equal
to nine.

We can then use this to solve the
second equation: negative three π¦ plus six π₯ equals 63. That gives us that negative three
π¦ plus six multiplied by nine equals 63. And we know that six multiplied by
nine is equal to 54. And then subtracting 54 from both
sides of this equation gives us that negative three π¦ is equal to nine.

Remember that our question asked us
to find the value of π₯ plus π¦ plus π§. And we now know the values of π₯,
π¦, and π§. So π₯ plus π¦ plus π§ equals nine
minus three add four. And that gives us 10.