Let a be a matrix of order `2xx2` such that `A^(2)=O`.
`A^(2)-(a+d)A+(ad-bc)I` is equal to

Options

A. `I`
B. `O`
C. `-I`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \( A^2 - (a + d)A + (ad - bc)I \) given that \( A^2 = O \) (the null matrix).

Let's denote the matrix \( A \) as follows:

\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

### Step 1: Calculate \( A^2 \)

We know that:

\[ A^2 = A \cdot A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

Calculating this gives:

\[ A^2 = \begin{pmatrix} a^2 + bc & ab + bd \\ ca + dc & cb + d^2 \end{pmatrix} \]

Since \( A^2 = O \), we have:

\[ \begin{pmatrix} a^2 + bc & ab + bd \\ ca + dc & cb + d^2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

This implies:

1. \( a^2 + bc = 0 \)
2. \( ab + bd = 0 \)
3. \( ca + dc = 0 \)
4. \( cb + d^2 = 0 \)

### Step 2: Substitute \( A^2 \) into the expression

Now we substitute \( A^2 \) into the expression:

\[ A^2 - (a + d)A + (ad - bc)I \]

Substituting \( A^2 = O \):

\[ O - (a + d)A + (ad - bc)I \]

This simplifies to:

\[ -(a + d)A + (ad - bc)I \]

### Step 3: Expand the expression

Now we expand \( -(a + d)A \):

\[ -(a + d)A = -(a + d) \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} -(a + d)a & -(a + d)b \\ -(a + d)c & -(a + d)d \end{pmatrix} \]

And for \( (ad - bc)I \):

\[ (ad - bc)I = (ad - bc) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} ad - bc & 0 \\ 0 & ad - bc \end{pmatrix} \]

### Step 4: Combine the two parts

Now we combine the two matrices:

\[ \begin{pmatrix} -(a + d)a + (ad - bc) & -(a + d)b \\ -(a + d)c & -(a + d)d + (ad - bc) \end{pmatrix} \]

### Step 5: Simplify the expression

Now we simplify each entry:

1. First entry:
\[ -(a + d)a + (ad - bc) = -a^2 - ad + ad - bc = -a^2 - bc \]

2. Second entry:
\[ -(a + d)b \]

3. Third entry:
\[ -(a + d)c \]

4. Fourth entry:
\[ -(a + d)d + (ad - bc) = -d^2 - ad + ad - bc = -d^2 - bc \]

Thus, we have:

\[ \begin{pmatrix} -a^2 - bc & -(a + d)b \\ -(a + d)c & -d^2 - bc \end{pmatrix} \]

### Final Result

Since \( A^2 = O \) implies \( a^2 + bc = 0 \) and \( d^2 + bc = 0 \), we can conclude that:

\[ A^2 - (a + d)A + (ad - bc)I = O \]