\(\begin{bmatrix} a && b \\ c && d \end{bmatrix}*\begin{bmatrix} a && b \\ c && d \end{bmatrix}=\begin{bmatrix} a^2+b*c &&a*b+b*d \\ c*a+d*c && d^2+c*b \end{bmatrix}= \begin{bmatrix} a^2+bc && b*(a+d) \\ c*(a+d) && d^2+bc \end{bmatrix}=\begin{bmatrix} 0 && 0 \\ 0 && 0 \end{bmatrix}\)

(The multiplication of the matrices is derived from the formula: \((A*B)i,j =\sum_{k=1}^{n}a(i,k)*b(k,j)\)

so we have a set of equations:

1. a²+bc=0

2. b*(a+d)=0

3. c*(a+d)=0

4. d²+bc=0

using the first and the fourth equations, we can find that :(a²+bc)-(d²+bc)=(0)-(0)=0=a²-d²=(a-d)*(a+d). therefore, we can divide our answer to two cases:

first case- a=d

that means b*(a+d)=0=b*(2a) and c*(a+d)=0=c*(2a). now we have a new set of equations:

1. a²+bc=0

2. b*(2a)=0

3. c*(2a)=0

we can make it simpler:

1. a²+bc=0

2. b*a=0

3. c*a=0

now we can divide that case to two different cases:

a=0:

now we have 1 equation:

1. a²+bc=0=0²+bc=bc=0. that means b or c must be equal to 0 (that or is a mathematical or: it means if both of them are 0 it works too)

that means the matrices that maintain the equations are of the following form:

\(\begin{bmatrix} 0 && b \\ c && 0 \end{bmatrix}\)(where c*b=0)

a is not 0:

that means we can divide the second and the third equations by a and get the set of equations:
 

1. a²+bc=0

2. b=0

3. c=0

that means a²+bc=a²+0*0=a². but that means a=0, and that doesnt make any sense because we know that a CANNOT be 0. So there is no solution

second case- d=-a

now we can derive a simpler set of equations:

1. a²+bc=0

2. b*(a-a)=0=b*0=0

3. c*(a-a)=0=c*0=0

now we have one equation- a²+bc=0:

a²+bc=0/ -a²= 

bc=-a²

that gives us the next set of matrices: \(\begin{bmatrix} (-1)^n\sqrt{-bc} && b \\ c && -(-1)^n\sqrt{-bc} \end{bmatrix} \)

(where n is a natural number, couldnt find the plus minus sign). but as you can see, every matrice from the first set of matrices is also a member of the second set of matrices, therefore we can simply say that every matrice that is a member of the second set is a solution (keep in mind that every matrice from that set is a solution)

but we dont want A=0 as a solution, so we simply have to add that b*c is not 0.

solution: \(\begin{bmatrix} (-1)^n\sqrt{-bc} && b \\ c && -(-1)^n\sqrt{-bc} \end{bmatrix} \)

(where b*c is not 0)

Matrix Problem: A \(\mathbf{\in \mu}\) (2x2).

Find the possible conditions and possible cases in the real numbers for which A² = 0

In linear algebra, a nilpotent matrix is a square matrix A such that

\({\displaystyle A^{k}=0\,}\)

for some positive integer k.

If A is a nilpotent matrix then det(A) = 0

\(\begin{array}{|rcll|} \hline det(A) &=& \begin{vmatrix} a & b \\ c&d \end{vmatrix} = ad-bc = 0 \\\\ \Rightarrow \qquad \mathbf{bc} & \mathbf{=} & \mathbf{ad} \\ \hline \end{array}\)

\(A^2 = 0 \\ \begin{array}{|rcll|} \hline A^2 &=& \begin{pmatrix} a & b \\ c&d \end{pmatrix} \cdot \begin{pmatrix} a & b \\ c&d \end{pmatrix} \\\\ &=& \begin{pmatrix} a^2+bc & b\cdot (a+d) \\ c\cdot (a+d) & d^2+bc \end{pmatrix} \quad & | \quad \mathbf{bc=ad} \\\\ &=& \begin{pmatrix} a^2+ad & b\cdot (a+d) \\ c\cdot (a+d) & d^2+ad \end{pmatrix} \\\\ &=& \begin{pmatrix} a\cdot (a+d) & b\cdot (a+d) \\ c\cdot (a+d) & d\cdot(a+d) \end{pmatrix} \\\\ A^2 &=& \underbrace{(a+d)}_{=0}\cdot \underbrace{\begin{pmatrix} a & b \\ c & d \end{pmatrix}}_{A\ne 0} = 0\\\\ a+d &=& 0 \\ \mathbf{a} &\mathbf{=}& \mathbf{-d} \\ \hline \end{array}\)

if \(A\ne 0 \) and \(A^2 = 0\), then \( a = -d\) and \(det(A) = 0\).

\(\text{Example } 1:\\ \begin{array}{|rcll|} \hline A &=& \begin{pmatrix} 0 & 1 \\ 0&0 \end{pmatrix} \\ det(A) &=& \begin{vmatrix} 0 & 1 \\ 0&0 \end{vmatrix} = 0\cdot 0 - 0 \cdot 1 = 0 \\ a &=& -d \\ 0 &=& 0\ \checkmark \\\\ A^2 &=& \begin{pmatrix} 0 & 1 \\ 0&0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 0&0 \end{pmatrix} = 0\ \checkmark \\ \hline \end{array}\)

\(\text{Example } 2:\\ \begin{array}{|rcll|} \hline A &=& \begin{pmatrix} 4 & 8 \\ -2 & -4 \end{pmatrix} \\ det(A) &=& \begin{vmatrix} 4 & 8 \\ -2 & -4 \end{vmatrix} = 4\cdot(-4) - (-2) \cdot 8 = -16+16 = 0 \\ a &=& -d \\ 4 &=& -(-4) \\ 4 &=&4\ \checkmark \\\\ A^2 &=& \begin{pmatrix} 4 & 8 \\ -2 & -4 \end{pmatrix} \cdot \begin{pmatrix} 4 & 8 \\ -2 & -4 \end{pmatrix} = 0\ \checkmark \\ \hline \end{array}\)