Let `A,B` and `C` be `nxxn` matrices. Which one of the following is a correct statement?

Options

A. If `A^(2)=O," then A=O"`
B. If `A^(3)+2A^(2)+3A+5O=O,` then A is invertible
C. If AB=AC then B=C
D. If AB=O, then A=O or B=O

Answer

To determine which statement is correct regarding the matrices \( A, B, \) and \( C \), we will analyze each statement one by one.

### Step 1: Analyze Statement A
**Statement A**: If \( A^2 = 0 \) (the null matrix), then \( A = 0 \).

To verify this, we start with the assumption that \( A^2 = 0 \). This means:
\[ A \cdot A = 0 \]
If \( A \) were invertible, we could multiply both sides by \( A^{-1} \):
\[ A^{-1} \cdot (A \cdot A) = A^{-1} \cdot 0 \implies I \cdot A = 0 \implies A = 0 \]
However, if \( A \) is not invertible, \( A^2 = 0 \) does not necessarily imply \( A = 0 \). For example, consider the matrix:
\[ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]
Here, \( A^2 = 0 \) but \( A \neq 0 \). Thus, **Statement A is not correct**.

### Step 2: Analyze Statement B
**Statement B**: If \( A^3 + 2A^2 + 3A + 5I = 0 \), then \( A \) is invertible.

We can rearrange this equation:
\[ A^3 + 2A^2 + 3A = -5I \]
If \( A \) were not invertible, then \( A \) would have a determinant of zero, leading to a contradiction since the right-hand side is a scalar multiple of the identity matrix (which is invertible). Thus, **Statement B is correct**.

### Step 3: Analyze Statement C
**Statement C**: If \( AB = AC \), then \( B = C \).

This statement is true only if \( A \) is invertible. If \( A \) is invertible, we can multiply both sides by \( A^{-1} \):
\[ A^{-1}AB = A^{-1}AC \implies IB = IC \implies B = C \]
However, if \( A \) is not invertible, \( AB = AC \) does not guarantee \( B = C \). For example, if \( A = 0 \), then \( AB = AC = 0 \) for any \( B \) and \( C \). Thus, **Statement C is not always correct**.

### Step 4: Analyze Statement D
**Statement D**: If \( AB = 0 \), then \( A = 0 \) or \( B = 0 \).

This statement is false in general. For instance, if \( A \) is a non-zero matrix and \( B \) is a non-zero matrix, it is possible for their product to be the zero matrix. For example:
\[ A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \]
Here, \( AB = 0 \) but neither \( A \) nor \( B \) is the zero matrix. Thus, **Statement D is not correct**.

### Conclusion
The only correct statement among the options is **Statement B**: If \( A^3 + 2A^2 + 3A + 5I = 0 \), then \( A \) is invertible.

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