Determinant and inverse matrix

Compute with Sarrus’ rule the determinant of

$$A=\left(\begin{array}{ccc}1& 3& 2\\ 2& -1& 1\\ 0& 4& 3\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute, for $a\in \mathbb{R}$, the determinant

$$\left|\begin{array}{cc}a+1& a\\ a& a-1\end{array}\right|.$$

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Compute the determinant of

$$A=\left(\begin{array}{cccc}2& 1& 0& 3\\ 0& 3& 0& 1\\ 1& -1& 2& 0\\ 0& 0& 0& 4\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute the determinant of

$$A=\left(\begin{array}{ccccc}1& 0& 0& 2& 0\\ 3& 2& 1& 0& 4\\ 0& 0& 3& 0& 0\\ 1& 0& 0& 1& 0\\ 2& 0& 5& 1& 1\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute the determinant of

$$A=\left(\begin{array}{cccc}1& -1& 2& 1\\ 2& -1& 3& 4\\ -1& 2& 1& 0\\ 3& -3& 5& 4\end{array}\right)$$

by reducing the matrix to triangular form.

Solution available to those who have unlocked the subject.

Compute the area of the triangle with vertices $P=(0,1)$, $Q=(4,2)$, $R=(1,5)$.

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Compute the inverse of

$$A=\left(\begin{array}{cc}4& 7\\ 1& 2\end{array}\right).$$

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Compute with cofactors the inverse of

$$A=\left(\begin{array}{ccc}1& 1& 0\\ 1& 2& 1\\ 0& 1& 3\end{array}\right).$$

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Determine for which values of $k\in \mathbb{R}$ the matrix

$$A_k=\left(\begin{array}{ccc}k& 1& 0\\ 1& k& 1\\ 0& 1& k\end{array}\right)$$

is invertible.

Solution available to those who have unlocked the subject.

Determine for which values of $k\in \mathbb{R}$ the matrix

$$A_k=\left(\begin{array}{cccc}k& 1& 0& 0\\ 1& k& 0& 0\\ 0& 0& k& 2\\ 0& 0& 2& k\end{array}\right)$$

is invertible.

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Let $A,B\in M_4(\mathbb{R})$ with $\det A=-2$ and $\det B=3$. Compute

$$\det \left(2A^{-1}{B}^T\right).$$

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Solve the matrix equation $XA=B$, with

$$A=\left(\begin{array}{cc}1& 1\\ 2& 3\end{array}\right),B=\left(\begin{array}{cc}1& 2\\ 1& 3\end{array}\right).$$

Solution available to those who have unlocked the subject.

Let $A\in M_n(\mathbb{R})$ be such that $A^{T}A=I_n$. Determine the possible values of $\det A$.

Solution available to those who have unlocked the subject.