§ 4

Let $A$ be a square matrix of order $n$. Express $\det \alpha A$ through $\det A$.

Let $A$ be a square matrix of order $2 n+1$, and $A^{T}=-A$. Prove that $\det A=0$.

Prove that the determinant of any triangular matrix equals the product of its diagonal entries.

Compute

One and the same number $t$ has been added to each entry of a matrix $A$. Prove that the determinant of the resulting matrix is a linear function of $t$.

Compute the determinant of order $n$ :

Two quadratic polynomials $a x^{2}+b x+c$ and $\alpha x^{2}+\beta x+\gamma$, $(a, \alpha \neq 0)$ have a common root. Prove that

Prove that the equation $X^{2}+E=O$ has no solutions among real matrices of odd order.

How many inversions are there in the permutation (5, 4, 3, 2, 1)?

Let a matrix $P$ of order $n$ have the form

where $A$ and $C$ are square submatrices of orders $k$ and $n-k$, and $O$ is a zero submatrix. By means of the complete expansion formula prove that $\det P=\det A \det C$.