Question

1. If $ X=\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right] $ and $ A=\left[\begin{array}{cc}a_{11} & 0 \\ 0 & a_{22}\end{array}\right] $. Find $ X^{T}(A X) $. Show that $ \left(X^{T} A\right) X=X^{T}(A X) $. 2. Show that $ A I=A $, given that $ A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right] $. Using matrix A , find $ M_{13}, M_{31}, M_{22}, C_{32}, C_{12}, C_{23} $ 3. Given that matrix $ C=\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ 2 a & 2 b & 2 c\end{array}\right] $, Find the $ |C| $. 4. The number of units of PIZZA sold at RACH heaven for the last 2 weeks are shown in the matrix A below, where the columns represent weeks and rows represent the two different types, "Something meat" and "Chicken Mushroom" \[ A=\left[\begin{array}{cc} 12 & 30 \\ 8 & 15 \end{array}\right] \] If each type sells for $ \$ 4 $, derive a matrix for the total revenue for RACH heaven over the two weeks period. 5. A company's input requirement over the next four weeks for the 3 inputs $ X, Y $ and $ Z $ are given (in number of units of each input) by the matrix $ R=\left[\begin{array}{cccc}2 & 0.5 & 1 & 7 \\ 6 & 3 & 8 & 2.5 \\ 4 & 5 & 2 & 0\end{array}\right] $. The company can buy these inputs from two suppliers, whose prices for these three inputs $ \mathrm{X}, \mathrm{Y} $ and Z are given (in K ) by the matrix $ P=\left[\begin{array}{lll}4 & 6 & 2 \\ 5 & 8 & 1\end{array}\right] $ where the two rows represent the suppliers and the three columns represent the input prices. Derive a matrix that will give the total input bill for the next four weeks for both suppliers.

          1. If \( X=\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right] \) and \( A=\left[\begin{array}{cc}a_{11} & 0 \\ 0 & a_{22}\end{array}\right] \). Find \( X^{T}(A X) \). Show that \( \left(X^{T} A\right) X=X^{T}(A X) \).
2. Show that \( A I=A \), given that \( A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right] \). Using matrix A , find \( M_{13}, M_{31}, M_{22}, C_{32}, C_{12}, C_{23} \)
3. Given that matrix \( C=\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ 2 a & 2 b & 2 c\end{array}\right] \), Find the \( |C| \).
4. The number of units of PIZZA sold at RACH heaven for the last 2 weeks are shown in the matrix A below, where the columns represent weeks and rows represent the two different types, "Something meat" and "Chicken Mushroom"
\[
A=\left[\begin{array}{cc}
12 & 30 \\
8 & 15
\end{array}\right]
\]

If each type sells for \( \$ 4 \), derive a matrix for the total revenue for RACH heaven over the two weeks period.
5. A company's input requirement over the next four weeks for the 3 inputs \( X, Y \) and \( Z \) are given (in number of units of each input) by the matrix \( R=\left[\begin{array}{cccc}2 & 0.5 & 1 & 7 \\ 6 & 3 & 8 & 2.5 \\ 4 & 5 & 2 & 0\end{array}\right] \). The company can buy these inputs from two suppliers, whose prices for these three inputs \( \mathrm{X}, \mathrm{Y} \) and Z are given (in K ) by the matrix \( P=\left[\begin{array}{lll}4 & 6 & 2 \\ 5 & 8 & 1\end{array}\right] \) where the two rows represent the suppliers and the three columns represent the input prices. Derive a matrix that will give the total input bill for the next four weeks for both suppliers.

1. If X=[
x1
x2
] and A=[
a11 0
0 a22
]. Find X^T(A X). Show that (X^T A) X=X^T(A X).
2. Show that A I=A, given that A=[
a11 a12 a13
a21 a22 a23
a31 a32 a33
]. Using matrix A , find M13, M31, M22, C32, C12, C23
3. Given that matrix C=[
a b c
d e f
2 a 2 b 2 c
], Find the |C|.
4. The number of units of PIZZA sold at RACH heaven for the last 2 weeks are shown in the matrix A below, where the columns represent weeks and rows represent the two different types, "Something meat" and "Chicken Mushroom"

A=[
12 30

8 15
]

If each type sells for $ 4, derive a matrix for the total revenue for RACH heaven over the two weeks period.
5. A company's input requirement over the next four weeks for the 3 inputs X, Y and Z are given (in number of units of each input) by the matrix R=[
2 0.5 1 7
6 3 8 2.5
4 5 2 0
]. The company can buy these inputs from two suppliers, whose prices for these three inputs X, Y and Z are given (in K ) by the matrix P=[
4 6 2
5 8 1
] where the two rows represent the suppliers and the three columns represent the input prices. Derive a matrix that will give the total input bill for the next four weeks for both suppliers.

Added by Stephen C.

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