Given matrices:
A = [1, 2; 4, 3]
B = [5; -3]
We need to compute the value of:
\[
\left[ \dfrac{{n(C) \left( |D|^2 + n(D) \right)}}{{n(A) - n(B)}} \right]
\]
where C = (AB)(B'A) and D = (B'A)(AB).
### Step 1: Compute the number of elements in matrices A and B
- A is a 2x2 matrix, so the number of elements in A, \( n(A) = 2 \times 2 = 4 \).
- B is a 2x1 matrix, so the number of elements in B, \( n(B) = 2 \times 1 = 2 \).
### Step 2: Find the product AB
Matrix A is 2x2 and matrix B is 2x1. Since the number of columns in A is equal to the number of rows in B, AB can be computed.
AB = A * B = [1, 2; 4, 3] * [5; -3]
Multiply:
AB = [(1*5 + 2*(-3)); (4*5 + 3*(-3))]
= [5 - 6; 20 - 9]
= [-1; 11]
So, AB is a 2x1 matrix.
### Step 3: Find the transpose of matrix B, B'
B' is the transpose of matrix B. Since B is 2x1, B' will be 1x2:
B' = [5, -3]
### Step 4: Find the product B'A
Now compute B'A, where B' is 1x2 and A is 2x2. The product B'A is:
B'A = [5, -3] * [1, 2; 4, 3]
Multiply:
B'A = [5*1 + (-3)*4, 5*2 + (-3)*3]
= [5 - 12, 10 - 9]
= [-7, 1]
So, B'A is a 1x2 matrix.
### Step 5: Compute matrix C and D
We have:
C = (AB)(B'A)
Since AB is a 2x1 matrix and B'A is a 1x2 matrix, the product C will be a 2x2 matrix.
C = [-1; 11] * [-7, 1]
Multiply:
C = [(-1)*(-7), (-1)*1; (11)*(-7), (11)*1]
= [7, -1; -77, 11]
So, C is a 2x2 matrix.
Now compute D:
D = (B'A)(AB)
Since B'A is a 1x2 matrix and AB is a 2x1 matrix, the product D will be a 1x1 matrix.
D = [-7, 1] * [-1; 11]
Multiply:
D = (-7)*(-1) + 1*11
= 7 + 11
= 18
So, D is a 1x1 matrix, and the value of D is 18.
### Step 6: Calculate \( n(C) \) and \( n(D) \)
- The number of elements in C, \( n(C) = 2 \times 2 = 4 \).
- The number of elements in D, \( n(D) = 1 \times 1 = 1 \).
### Step 7: Calculate \( |D|^2 \)
Since D is a 1x1 matrix, \( |D| = D = 18 \).
Thus:
\( |D|^2 = 18^2 = 324 \).
### Step 8: Final Calculation
Now, we can substitute all the values into the given expression:
\[
\left[ \dfrac{{n(C) \left( |D|^2 + n(D) \right)}}{{n(A) - n(B)}} \right]
= \dfrac{{4 \times (324 + 1)}}{{4 - 2}}
= \dfrac{{4 \times 325}}{{2}}
= \dfrac{{1300}}{{2}} = 650
\]
Thus, the value of the expression is **650**.
