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Question 1 - Orthogonal

2 \times 2

matrices (a) Draw a picture In the grid pr...

May 12, 2024

Solution by Steps

step 1

To find the product of the given matrices, we first need to compute the transpose of the vector

\{0,1,0\}

and the transpose of the matrix

\left\{\begin{array}{ccc}1 &amp; 2 &amp; 3\\4 &amp; 5 &amp; 6\end{array}\right\}

step 2

The transpose of the vector

\{0,1,0\}

is a row matrix

(0, 1, 0)

step 3

The transpose of the matrix

\left\{\begin{array}{ccc}1 &amp; 2 &amp; 3\\4 &amp; 5 &amp; 6\end{array}\right\}

\left\{\begin{array}{cc}1 &amp; 4\\2 &amp; 5\\3 &amp; 6\end{array}\right\}

step 4

We then subtract the transpose of the matrix

\left\{\begin{array}{ccc}1 &amp; 0 &amp; 1\\0 &amp; 1 &amp; 0\end{array}\right\}

, which is

\left\{\begin{array}{cc}1 &amp; 0\\0 &amp; 1\\1 &amp; 0\end{array}\right\}

, from the previously obtained transpose

step 5

The result of the subtraction is

\left\{\begin{array}{cc}1 &amp; 4\\2 &amp; 5\\3 &amp; 6\end{array}\right\} - \left\{\begin{array}{cc}1 &amp; 0\\0 &amp; 1\\1 &amp; 0\end{array}\right\} = \left\{\begin{array}{cc}0 &amp; 4\\2 &amp; 4\\2 &amp; 6\end{array}\right\}

step 6

Now we multiply the row matrix

(0, 1, 0)

by the resulting matrix from step 5 to get

(0, 1, 0) \cdot \left\{\begin{array}{cc}0 &amp; 4\\2 &amp; 4\\2 &amp; 6\end{array}\right\} = (2, 4)

step 7

Finally, we multiply the resulting row matrix

(2, 4)

by the matrix

\left\{\begin{array}{cc}1 &amp; 2\\0 &amp; 6\end{array}\right\}

to get

(2, 4) \cdot \left\{\begin{array}{cc}1 &amp; 2\\0 &amp; 6\end{array}\right\} = (2, 26)

1 Answer

The final result of the matrix operation is

(2, 26)

Key Concept

Matrix multiplication and the properties of transpose operations are essential in solving this problem.

Explanation

The solution involves transposing vectors and matrices, performing matrix subtraction, and then carrying out matrix multiplication in the correct order, as matrix multiplication is not commutative.