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Evaluate : if possible (i) [3,2] [{:(,...

Evaluate : if possible
(i) `[3,2] [{:(,2),(,0):}]` (ii) `[1 , -2] [{:(,-2,3),(,-1,4):}]`
(iii) `[{:(,6,4),(,3,-1):}] [{:(,-1),(,3):}]` (iv) `[{:(,6,4),(,3,-1):}] [{:(,-1),(,3):}]`

Text Solution

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The correct Answer is:
Let's evaluate each part step by step. ### Part (i): Evaluate `[3, 2]` and `[{(2),(0)}]` 1. Identify the dimensions of the matrices: - The first matrix `[3, 2]` is a 1x2 matrix. - The second matrix `[{(2),(0)}]` is a 2x1 matrix. 2. Check if multiplication is possible: - The number of columns in the first matrix (2) matches the number of rows in the second matrix (2). Therefore, multiplication is possible. 3. Perform the multiplication: \[ [3, 2] \cdot \begin{bmatrix} 2 \\ 0 \end{bmatrix} = (3 \cdot 2) + (2 \cdot 0) = 6 + 0 = 6 \] ### Part (ii): Evaluate `[1, -2]` and `[{(-2, 3), (-1, 4)}]` 1. Identify the dimensions of the matrices: - The first matrix `[1, -2]` is a 1x2 matrix. - The second matrix `[{(-2, 3), (-1, 4)}]` is a 2x2 matrix. 2. Check if multiplication is possible: - The number of columns in the first matrix (2) matches the number of rows in the second matrix (2). Therefore, multiplication is possible. 3. Perform the multiplication: \[ [1, -2] \cdot \begin{bmatrix} -2 & 3 \\ -1 & 4 \end{bmatrix} = (1 \cdot -2 + -2 \cdot -1, 1 \cdot 3 + -2 \cdot 4) \] \[ = (-2 + 2, 3 - 8) = (0, -5) \] ### Part (iii): Evaluate `[{(6, 4), (3, -1)}]` and `[{(-1), (3)}]` 1. Identify the dimensions of the matrices: - The first matrix `[{(6, 4), (3, -1)}]` is a 2x2 matrix. - The second matrix `[{(-1), (3)}]` is a 2x1 matrix. 2. Check if multiplication is possible: - The number of columns in the first matrix (2) matches the number of rows in the second matrix (2). Therefore, multiplication is possible. 3. Perform the multiplication: \[ \begin{bmatrix} 6 & 4 \\ 3 & -1 \end{bmatrix} \cdot \begin{bmatrix} -1 \\ 3 \end{bmatrix} = (6 \cdot -1 + 4 \cdot 3, 3 \cdot -1 + -1 \cdot 3) \] \[ = (-6 + 12, -3 - 3) = (6, -6) \] ### Part (iv): Evaluate `[{(6, 4), (3, -1)}]` and `[{(-1), (3)}]` This part is identical to part (iii), so the evaluation will yield the same result. 1. The result is: \[ (6, -6) \] ### Summary of Results: - Part (i): 6 - Part (ii): (0, -5) - Part (iii): (6, -6) - Part (iv): (6, -6)
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