Video: Subtracting 2 × 2 Matrices

[6, −4 and −3, 2] − [5, −3 and −4, 1]^𝑡 = _. [A] [1, −8 and −6, 1] [B] [11, 0 and 0, 3] [C] 𝐼 [D] 𝑂

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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).

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