Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular, without using $\lambda_1\cdot\ldots\cdot\lambda_n = det(A)$.
41932
Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular.
106398
If $A$ is invertible show that $\det(A) \neq 0$
6514
Prove $T$ has at most two distinct eigenvalues
14320
How to prove that eigenvectors from different eigenvalues are linearly independent
153448
Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$.
104293
How to prove "eigenvalues of polynomial of matrix $A$ = polynomial of eigenvalues of matrix $A$ "
158724
Show that $B$ is diagonalizable if If $AB=BA$ and $A$ has distinct real eigenvalues
73677
If the matrix $A^3 = 0$, is $A$ singular?
152932
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
104424
If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$
68132
Small positive eigenvalues found for a negative definite matrix
99600
Show that a 2x2 matrix A is symmetric positive definite if and only if A is symmetric, trace(A) > 0 and det(A) > 0
18297
Suppose that each of the row sums of an $n\times n$ matrix $A$ is equal to zero. Show that $A$ must be singular.
9220
If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$
49158
Multiplicity of eigenvalues
37537
Show that $det(A)=0$ for a Matrix $A$ Such That for $X=[2,3,4]^{T}$, $AX=0$
136643
Continuity of eigenvalues and spectral radius for a general matrix