The results of given matrices can be obtained using matrix multiplication.
<h3>Find the results of the given matrices:</h3>
Here in the question it is given that,
A =
, B =
, C =
, D =
,
E =
, F = ![\left[\begin{array}{ccc}i&1-3i\\0&4+i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%261-3i%5C%5C0%264%2Bi%5Cend%7Barray%7D%5Cright%5D)
We have to find AB, BC, CA, CD,
, F²,
,
and FE.
- AB =
![\left[\begin{array}{ccc}1&-1&2\\3&1&4\end{array}\right]\left[\begin{array}{ccc}2&-1&3\\5&1&2\\4&6&-2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26-1%262%5C%5C3%261%264%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-1%263%5C%5C5%261%262%5C%5C4%266%26-2%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 1×2 + (-1)×5 + 2×4 = 5, a₁₂ = 1×(-1) + (-1)×1 + 2×6 = 10, a₁₃ = 1×3 + (-1)×2 + 2×(-2) = -3, a₂₁ = 3×2 + 1×5 + 4×4 = 27, a₂₂ = 3×(-1) + 1×1 + 4×6 = 22, a₂₃ = 3×3 + 1×2 + 4×(-2) = 3
AB =
- BC =
![\left[\begin{array}{ccc}1\\-1\\2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%5C%5C-1%5C%5C2%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 2×1 + (-1)×(-1) + 3×2 = 9, a₂₁ = 5×1 + 1×(-1) + 2×2 = 8, a₃₁ = 4×1 + 6×(-1) + (-2)×2 = -6
BC = ![\left[\begin{array}{ccc}9\\8\\-6\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C8%5C%5C-6%5Cend%7Barray%7D%5Cright%5D)
- CA, CA is not defined since dimension of the matrices are 3×1 and 2×3
a₁₁ = 1×(2-i) + 3×(-i) = 2-4i, a₁₂ = 1x(1+i) + 3×(2+4i) = 7+13i, a₂₁ = -1×(2-i) + 1×(-i) = -2, a₂₂ = -1×(1+i) + 1×(2+4i) = 1+3i, a₃₁ = 2×(2-i) + 4×(-i) = 4-6i, a₃₂ = 2×(1+i) + 4×(2+4i) = 10+18i
![A^{T}E = \left[\begin{array}{ccc}2-4i&7+13i\\-2&1+3i\\4-6i&10+18i\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7BT%7DE%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2-4i%267%2B13i%5C%5C-2%261%2B3i%5C%5C4-6i%2610%2B18i%5Cend%7Barray%7D%5Cright%5D)
- CD =
![\left[\begin{array}{ccc}2&-2&3\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-2%263%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 1×2 = 2, a₁₂ = 1×(-2) = -2, a₁₃ = 1×3 = 3, a₂₁ = -1×2 = -2, a₂₂ = -1×(-2) = 2, a₂₃ = -1×3 = -3,a₃₁= 2×2 = 4, a₃₂ = 2×(-2) = -4, a₃₃ = 2×3 = 6
CD = ![\left[\begin{array}{ccc}2&-2&3\\-2&2&-3\\4&-4&6\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-2%263%5C%5C-2%262%26-3%5C%5C4%26-4%266%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 1×1 + (-1)×(-1) + 2×2 = 6, a₁₂ = 1×3 + (-1)×1 + 2×4 = 10
![C^{T}A^{T}=\left[\begin{array}{ccc}6&10\end{array}\right]](https://tex.z-dn.net/?f=C%5E%7BT%7DA%5E%7BT%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D6%2610%5Cend%7Barray%7D%5Cright%5D)
- F² =
![\left[\begin{array}{ccc}i&1-3i\\0&4+i\end{array}\right]\left[\begin{array}{ccc}i&1-3i\\0&4+i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%261-3i%5C%5C0%264%2Bi%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%261-3i%5C%5C0%264%2Bi%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = i×i + (1-3i)×0 = -1,a₁₂ = i×(1-3i) + (1-3i)×(4+i) = 10-10i, a₂₁= 0×i + (4+i)×0 = 0, a₂₂ = 0×(1-3i) + (4+i)×(4+i) = 15+8i
F² = ![\left[\begin{array}{ccc}-1&10-10i\\0&15+8i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%2610-10i%5C%5C0%2615%2B8i%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 2×2 + (-1)×(-2) + 3×3 = 15, a₂₁ = 5×2 + 1×(-2) + 2×3 = 14, a₃₁ = 4×2 + 6×(-2) + (-2)×3 = -10
![BD^{T}= \left[\begin{array}{ccc}15\\14\\-10\end{array}\right]](https://tex.z-dn.net/?f=BD%5E%7BT%7D%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D15%5C%5C14%5C%5C-10%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = 1×1 + 3×3 = 10, a₁₂ = 1×(-1) + 3×1 = 2, a₁₃ = 1×2 + 3×4 = 14, a₂₁ = -1×1 + 1×3 = 2, a₂₂ = -1×(-1) + 1×1 = 2, a₂₃ = -1×2 + 1×4 = 2, a₃₁ = 2×1 + 4×3 = 14, a₃₂ = 2×(-1) + 4×1 = 2, a₃₃ = 2×2 + 4×4 = 20
![A^{T} A=\left[\begin{array}{ccc}10&2&14\\2&2&2\\14&2&20\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7BT%7D%20A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D10%262%2614%5C%5C2%262%262%5C%5C14%262%2620%5Cend%7Barray%7D%5Cright%5D)
- FE =
![\left[\begin{array}{ccc}2-i&1+i\\-i&2+4i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2-i%261%2Bi%5C%5C-i%262%2B4i%5Cend%7Barray%7D%5Cright%5D)
a₁₁ = i×(2-i) + (1-3i)×(-i) = -2+i, a₁₂ = i×(1+i) + (1-3i)×(2+4i) = 13-i, a₂₁ = 0×(2-i) + (4+i)×(-i) = 1-4i, a₂₂ = 0×(1+i) + (4+i)×(2+4i) = 4+18i
FE = ![\left[\begin{array}{ccc}-2+i&13-i\\1-4i&4+18i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%2Bi%2613-i%5C%5C1-4i%264%2B18i%5Cend%7Barray%7D%5Cright%5D)
Hence we can obtain the results of the required matrices using matrix multiplication.
Disclaimer: The question was given incomplete on the portal. Here is the complete question.
Question: Let A =
, B =
, C =
, D =
, E =
, F = ![\left[\begin{array}{ccc}i&1-3i\\0&4+i\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%261-3i%5C%5C0%264%2Bi%5Cend%7Barray%7D%5Cright%5D)
For each item, decide whether or not the given expression is defined. for each item that is defined, compute the result.
AB, BC, CA, CD,
, F²,
,
and FE
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