Video Transcript
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).
