Let the matrix `A=[(1,2,3),(0, 1,2),(0,0,1)] and BA=A` where B represent `3xx3` order matrix. If the total number of 1 in matrix `A^(-1)` and matrix B are p and q respectively. Then the value of `p+q` is equal to

Options

A. 3
B. 4
C. 5
D. 7

Answer

To solve the problem, we need to find the inverse of the matrix \( A \) and then count the number of 1's in both \( A^{-1} \) and matrix \( B \), where \( B \) is defined by the equation \( BA = A \).

### Step 1: Define the matrix \( A \)
The given matrix \( A \) is:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Find the determinant of matrix \( A \)
To find the inverse, we first need to calculate the determinant of \( A \).

Using the formula for the determinant of a \( 3 \times 3 \) matrix:
\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]
where \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).

For our matrix \( A \):
- \( a = 1, b = 2, c = 3 \)
- \( d = 0, e = 1, f = 2 \)
- \( g = 0, h = 0, i = 1 \)

Calculating the determinant:
\[ \text{det}(A) = 1(1 \cdot 1 - 2 \cdot 0) - 2(0 \cdot 1 - 2 \cdot 0) + 3(0 \cdot 0 - 1 \cdot 0) = 1(1) - 2(0) + 3(0) = 1 \]

### Step 3: Find the adjoint of matrix \( A \)
The adjoint of a matrix is obtained by taking the cofactor of each element and transposing the resulting matrix.

Calculating the cofactors:
- For \( a_{11} = 1 \): Minor is \( \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \) → determinant = 1 → cofactor = 1
- For \( a_{12} = 2 \): Minor is \( \begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} \) → determinant = 0 → cofactor = -0 = 0
- For \( a_{13} = 3 \): Minor is \( \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \) → determinant = 0 → cofactor = 0

Continuing this process for all elements, we find:
\[ \text{adj}(A) = \begin{pmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 4: Calculate the inverse of matrix \( A \)
The inverse of a matrix is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Since \( \text{det}(A) = 1 \):
\[ A^{-1} = \begin{pmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 5: Count the number of 1's in \( A^{-1} \)
Counting the number of 1's in \( A^{-1} \):
- In \( A^{-1} \), we have 3 ones.

Thus, \( p = 3 \).

### Step 6: Determine matrix \( B \)
From the equation \( BA = A \), we deduce that \( B \) must be the identity matrix \( I \) since multiplying \( A \) by the identity matrix yields \( A \):
\[ B = I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 7: Count the number of 1's in matrix \( B \)
Counting the number of 1's in \( B \):
- In \( B \), we have 3 ones.

Thus, \( q = 3 \).

### Step 8: Calculate \( p + q \)
Finally, we calculate:
\[ p + q = 3 + 3 = 6 \]

### Conclusion
The value of \( p + q \) is \( 6 \).