[Math] When does trace and determinant of a 2 x 2 matrix equal each other? (Linear Algebra)

determinantlinear algebramatricestrace

  • Background Information:

I am new to linear algebra, and I recently came across this homework question that I am confused about. I appreciate any explanation that can help me improve my solution.

  • Question:

What condition on the entries of a 2×2 matrix A means Tr(A) = det(A)? Provide
two distinct examples of 2×2 matrices which satisfy this.

  • My approach (Not Complete):

Considering the following 2 x 2 matrix, the det(A) = 4, and Tr(A) = 4

\begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix}

However, considering this 2 x 2 matrix, the det(A) = 9, and Tr(A) = 6

\begin{bmatrix}
3 & 0\\
0 & 3
\end{bmatrix}

I think the condition would be having 2 x 2 matrix such that the matrix is
(symmetric) and (n = 2).

\begin{bmatrix}
n & 0\\
0 & n
\end{bmatrix}

My solution makes sense, but I feel it is incomplete. Am I missing a key point or a concept that I can add to my answer?

  • Edited:

I have tried this solution with so many numbers and it seems to work. Would this be an acceptable solution?

\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}

such that a = c = d and b = c – 2, so here is an example

\begin{bmatrix}
5 & 3\\
5 & 5
\end{bmatrix}

det(A) = 25 – 15 = 10 , and Tr(A) = 5 + 5 = 10

Best Answer

consider below matrix $$M= \begin{bmatrix} a & b \\ c & d\\ \end{bmatrix} $$ the trace will be: $$Tr(M)=a+d$$

and the determinant : $$det(m)=(ad-bc)$$

then according to your problem:

$$a+d=ad-bc$$

so chose a and d arbitrary and then chose b and c in the way that the above equation will hold. for example:

$$a=10,d=20 $$

bc=170 and you can choose: $$ b=17 ,c=10$$

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