Consider matrix `A=[a_(ij)]_(nxxn)`. Form the matrix `A-lamdal, lamda` being a number, real of complex.
`A-lamdal=[{:(a_11-lamda,a_12,...,a_(1n)),(s_21,a_22-lamda,...,a_(2n)),(...,...,...,...),(a_(n1),a_(n2),...,a_(n n)-lamda):}]`
Then det `(A-lamdaI)=(-1)^n[lamda^n+b_1lamda^(n-1)+b_2lamda^(n-2)+...+b_(n)]`.
An important rheorem tells us that the matrix A satisfies the equation `X^n+b_1x^(n-1)+b_2x^(n-2)+...+b_2=0.` This equation is called hte characteristic equation of A. For all the questions on theis passeage, take `A=[{:(1,4),(2,3):}]`
The matrix A satisfies the matrix equation

Options

A. (a)`A^2-4A-5l=0`
B. (b)`A^2-4A-5=0`
C. (c)`A^2+5A-5l=0`
D. (d)`A^2+4A-5=0`

Answer

To solve the problem step by step, we will follow the procedure outlined in the video transcript.

### Step 1: Define the Matrix A
Given the matrix \( A \):
\[ A = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \]

### Step 2: Form the Matrix \( A - \lambda I \)
The identity matrix \( I \) for a \( 2 \times 2 \) matrix is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
Thus, \( \lambda I \) is:
\[ \lambda I = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} \]
Now, we compute \( A - \lambda I \):
\[ A - \lambda I = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} - \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} = \begin{pmatrix} 1 - \lambda & 4 \\ 2 & 3 - \lambda \end{pmatrix} \]

### Step 3: Compute the Determinant
Next, we need to find the determinant of the matrix \( A - \lambda I \):
\[ \text{det}(A - \lambda I) = (1 - \lambda)(3 - \lambda) - (4)(2) \]
Calculating this gives:
\[ = (1 - \lambda)(3 - \lambda) - 8 \]
Expanding the determinant:
\[ = 3 - \lambda - 3\lambda + \lambda^2 - 8 \]
\[ = \lambda^2 - 4\lambda - 5 \]

### Step 4: Set the Characteristic Equation
The characteristic equation is set by equating the determinant to zero:
\[ \lambda^2 - 4\lambda - 5 = 0 \]

### Step 5: Write the Matrix Equation
We can express this characteristic equation in terms of the matrix \( A \):
\[ A^2 - 4A - 5I = 0 \]
where \( I \) is the identity matrix.

### Final Result
Thus, the matrix \( A \) satisfies the equation:
\[ A^2 - 4A - 5I = 0 \]