1. Multiply the following matrices: (a) eginbmatrix a b...

1. Multiply the following matrices: (a) egin{bmatrix} a & b \ b & a end{bmatrix} egin{bmatrix} a & -b \ -b & a end{bmatrix} (b) egin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{bmatrix} egin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 end{bmatrix} (The first matrix here is called the identity matrix. Can you guess why?) (c) egin{bmatrix} vec{u_1}^T \ vec{u_2}^T \ vec{u_3}^T end{bmatrix} egin{bmatrix} vec{u_1} & vec{u_2} & vec{u_3} end{bmatrix} where vec{u_1}, vec{u_2}, and vec{u_3} are a set of orthonormal vectors.

Transcript

00:01 In the question we have given that the matrix a which is given as negative 1, negative 5, 8 as a 1 0, 0 as a 2nd row, then 3rd row contain the element 0, 1 and 4.

00:19 Also we have given the matrix b which is defined as the 1st row 0, 5, 3, then negative 1, 3 and 2.

00:29 Then we have given the matrix c is equals to 2 -1 -negative 1 -negative 1.

00:37 Also, the metric d is given as 2 -0 -negative 1, 3, then 1 and negative 2.

00:48 Now in this question we have to calculate the value of a times b transpose.

00:55 So from the given matrix we have the matrix a is negative.

00:59 1, 5, 8, 0, 0, 0, 0, 4.

01:06 Now we need to take the transpose of matrix d, that is, matrix b is given as 0, 5, 3, negative 1, 3, sorry, negative 1, 3, 2, transpose, that is we have negative 1, 5, 8, 0, 0 ,0, 0 ,000, negative 1 ,000, matrix a and b transpose can be written as 0, negative 1, 5, 3, 3 and 2.

01:37 We simplifying use the value of a times b transpose is equals to negative 1, 2, negative 3, negative to 17 and 11 which is a product of 2 matrix a times b transpose.

01:57 Now in the second part we have to calculate the value of 2 times d transpose minus 3.

02:06 So we take 2 times d transpose that is 0 to 0 negative 1 3 1 negative 2's transpose.

02:18 Simplifying it we get.

02:20 Sorry here we have to calculate 2 times d transpose minus 3 times b so minus 3 times we.

02:27 Matrix b is given as 0, 5, 3, negative 1, 3 and 2.

02:33 If we simplify it, we get 2 times b transpose, sorry d transpose it will be 2 negative 1, 1, 0, 3, 2...

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