Finding matrices that square to the zero matrix, denoted by $0$, is a fascinating problem in linear algebra. This exploration delves into the realm of nilpotent matrices, which are matrices whose powers eventually become the zero matrix. We will determine all $2\times2$ matrices $A$ such that $A^2=0$.
Let $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ be a $2\times2$ matrix. We want to find the conditions on $a$, $b$, $c$, and $d$ such that $A^2 = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$. Computing $A^2$, we have:
$A^2 = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{bmatrix}.$
Therefore, we need the following equations to hold:
\begin{align*} a^2 + bc &= 0, \ ab + bd &= 0, \ ac + cd &= 0, \ bc + d^2 &= 0. \end{align*}
Analyzing the Equations
Let's analyze these equations to find the solutions for $a$, $b$, $c$, and $d$.
From the second equation, $ab + bd = 0$, we can factor out a $b$ to get $b(a + d) = 0$. This tells us either $b = 0$ or $a + d = 0$.
Similarly, from the third equation, $ac + cd = 0$, we can factor out a $c$ to get $c(a + d) = 0$. This tells us either $c = 0$ or $a + d = 0$.
Now, let's consider the different cases:
Case 1: $b = 0$ and $c = 0$
If $b = 0$ and $c = 0$, the first and fourth equations become $a^2 = 0$ and $d^2 = 0$, respectively. This means $a = 0$ and $d = 0$.
Therefore, in this case, the matrix $A$ is of the form:
$A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}.$
Case 2: $b = 0$ and $a + d = 0$
If $b = 0$ and $a + d = 0$, the first and fourth equations become $a^2 = 0$ and $d^2 = 0$, respectively. This means $a = 0$ and $d = 0$.
Therefore, in this case, the matrix $A$ is of the form:
$A = \begin{bmatrix} 0 & 0 \ c & 0 \end{bmatrix},$
where $c$ can be any real number.
Case 3: $c = 0$ and $a + d = 0$
If $c = 0$ and $a + d = 0$, the first and fourth equations become $a^2 = 0$ and $d^2 = 0$, respectively. This means $a = 0$ and $d = 0$.
Therefore, in this case, the matrix $A$ is of the form:
$A = \begin{bmatrix} 0 & b \ 0 & 0 \end{bmatrix},$
where $b$ can be any real number.
Case 4: $a + d = 0$
If $a + d = 0$, then $d = -a$. Substituting into the first and fourth equations, we get:
\begin{align*} a^2 + bc &= 0, \ bc + a^2 &= 0. \end{align*}
These equations are identical, so we only need to consider one. Solving for $c$, we get $c = -\frac{a^2}{b}$ (assuming $b \neq 0$).
Therefore, in this case, the matrix $A$ is of the form:
$A = \begin{bmatrix} a & b \ -\frac{a^2}{b} & -a \end{bmatrix},$
where $a$ and $b$ can be any real numbers with $b \neq 0$.
Summary of Solutions
In summary, we have found the following solutions for the $2 \times 2$ matrices $A$ that satisfy $A^2 = 0$:
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The zero matrix: $A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$
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Matrices with a single nonzero entry: $A = \begin{bmatrix} 0 & 0 \ c & 0 \end{bmatrix}$ or $A = \begin{bmatrix} 0 & b \ 0 & 0 \end{bmatrix}$, where $b$ and $c$ are any real numbers.
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Matrices with a specific relationship between entries: $A = \begin{bmatrix} a & b \ -\frac{a^2}{b} & -a \end{bmatrix}$, where $a$ and $b$ are any real numbers with $b \neq 0$.
Geometric Interpretation
These solutions have interesting geometric interpretations. The zero matrix represents a transformation that maps all vectors to the origin. The matrices with a single nonzero entry represent transformations that project vectors onto a line (either the $x$-axis or the $y$-axis). The last type of matrix represents a shear transformation, which shifts points horizontally or vertically depending on their position.
Nilpotency
The matrices we found are examples of nilpotent matrices. A matrix is nilpotent if some power of the matrix equals the zero matrix. In our case, we found all the $2 \times 2$ matrices that are nilpotent of index 2 (meaning their square is the zero matrix).
In higher dimensions, finding all nilpotent matrices becomes more complex. However, the general idea is to use a similar approach of analyzing the equations that arise from setting the powers of the matrix equal to the zero matrix.
By exploring the properties of nilpotent matrices, we gain valuable insights into the structure and behavior of linear transformations. This understanding finds applications in various fields such as physics, engineering, and computer science.
