IB Mathematics Applications & Interpretation Question Type 6: Using Technology to Find Inverse and Determinants for Nxn Matrices Where n > 2 Exercises

Use technology to find the inverse of the matrix $$N=\left(\begin{array}{cccc}1& 2& 0& 0\\ 0& 1& 3& 0\\ 0& 0& 1& 4\\ 0& 0& 0& 1\end{array}\right).$$

Compute $\det (AB)$ without forming $AB$ explicitly.

Compute the determinant of the following $4\times 4$ matrix: $$M=\left(\begin{array}{cccc}3& 0& 2& -1\\ 1& 2& 0& 4\\ 0& 0& 1& 5\\ 2& -1& 3& 0\end{array}\right)$$

Compute the determinant of the following $3\times 3$ matrix: $$\left(\begin{array}{ccc}1& 2& 3\\ 4& 0& -1\\ 2& 5& 2\end{array}\right)$$

Compute the determinant of the matrix with fractional entries: $$Q=\left(\begin{array}{ccc}\frac{1}{2}& 2& 0\\ -1& \frac{3}{4}& 1\\ 2& -1& \frac{1}{3}\end{array}\right).$$

Determine whether the following matrix is invertible. If it is, find its inverse; if not, explain why not: $$B=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)$$

Find the inverse of the following $5\times 5$ diagonal matrix: $$D=\mathrm{diag}(2,3,4,5,6).$$

Consider the following $4\times 4$ matrix: $$R=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 6& 8\\ 1& 0& -1& -2\\ 0& 1& 1& 1\end{array}\right).$$ Compute its determinant and state whether it is invertible.

Use technology to find the determinant and the inverse of the matrix $$P=\left(\begin{array}{cccc}2& -1& 0& 3\\ 1& 0& 4& -2\\ 0& 5& 1& 1\\ 3& 0& -1& 2\end{array}\right).$$

Let $$A=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right).$$ Use technology to find $A^{-1}$ and compute $\det (A)$.

Find the inverse of the matrix $$A=\left(\begin{array}{ccc}2& 1& 0\\ -1& 1& 3\\ 0& 2& 1\end{array}\right)$$