If `A=[[1,3] , [3,4]]` and `A^2-kA-5I_2=0` then `k=`

Options

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

Answer

To solve the problem, we need to find the value of \( k \) in the equation \( A^2 - kA - 5I_2 = 0 \) given the matrix \( A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} \).

### Step 1: Calculate \( A^2 \)
To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} \]

Calculating the elements of \( A^2 \):

- First row, first column:
\[ 1 \cdot 1 + 3 \cdot 3 = 1 + 9 = 10 \]

- First row, second column:
\[ 1 \cdot 3 + 3 \cdot 4 = 3 + 12 = 15 \]

- Second row, first column:
\[ 3 \cdot 1 + 4 \cdot 3 = 3 + 12 = 15 \]

- Second row, second column:
\[ 3 \cdot 3 + 4 \cdot 4 = 9 + 16 = 25 \]

Thus, we have:
\[ A^2 = \begin{bmatrix} 10 & 15 \\ 15 & 25 \end{bmatrix} \]

### Step 2: Define \( I_2 \)
The identity matrix \( I_2 \) of order 2 is:
\[ I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

### Step 3: Calculate \( 5I_2 \)
Now, we multiply \( I_2 \) by 5:
\[ 5I_2 = 5 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \]

### Step 4: Substitute into the equation
We substitute \( A^2 \), \( kA \), and \( 5I_2 \) into the equation \( A^2 - kA - 5I_2 = 0 \):

\[ \begin{bmatrix} 10 & 15 \\ 15 & 25 \end{bmatrix} - k \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} - \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} = 0 \]

This simplifies to:
\[ \begin{bmatrix} 10 - k - 5 & 15 - 3k \\ 15 - 3k & 25 - 4k - 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

### Step 5: Set up equations
From the matrix equation, we set up the following equations:

1. \( 10 - k - 5 = 0 \)
2. \( 15 - 3k = 0 \)
3. \( 25 - 4k - 5 = 0 \)

### Step 6: Solve the equations
1. From the first equation:
\[ 10 - k - 5 = 0 \implies k = 5 \]

2. From the second equation:
\[ 15 - 3k = 0 \implies 3k = 15 \implies k = 5 \]

3. From the third equation:
\[ 25 - 4k - 5 = 0 \implies 20 - 4k = 0 \implies 4k = 20 \implies k = 5 \]

All equations give us \( k = 5 \).

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{5} \]