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Exercises for Question Type 6: Using technology to find inverse and determinants for nxn matrices where n > 2 - IB | RevisionDojo

Question 1

Use technology to find the inverse of the matrix

N = \begin{pmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 3 & 0\\ 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 1 \end{pmatrix}.

[2]

Question 2

Let

A = \begin{pmatrix} 1 & 2 & 1\\ 0 & 3 & 4\\ -1 & 2 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 & 1\\ 1 & -1 & 3\\ 0 & 4 & -2 \end{pmatrix}.

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

[4]

Question 3

Compute the determinant of the following $4 \times 4$ matrix: $M = \begin{pmatrix} 3 & 0 & 2 & -1 \\ 1 & 2 & 0 & 4 \\ 0 & 0 & 1 & 5 \\ 2 & -1 & 3 & 0 \end{pmatrix}$

[5]

Question 4

Compute the determinant of the following $3\times3$ matrix:

\begin{pmatrix} 1 & 2 & 3 \\ 4 & 0 & -1 \\ 2 & 5 & 2 \end{pmatrix}

[4]

Question 5

Compute the determinant of the matrix with fractional entries:

Q = \begin{pmatrix} \frac{1}{2} & 2 & 0\\ -1 & \frac{3}{4} & 1\\ 2 & -1 & \frac{1}{3} \end{pmatrix}.

[4]

Question 6

Determine whether the following matrix is invertible. If it is, find its inverse; if not, explain why not:

B = \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}

[4]

Question 7

Find the inverse of the following $5\times5$ diagonal matrix:

D = \operatorname{diag}(2,3,4,5,6).

[2]

Question 8

Consider the following $4\times4$ matrix:

R = \begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 4 & 6 & 8\\ 1 & 0 & -1 & -2\\ 0 & 1 & 1 & 1 \end{pmatrix}.

Compute its determinant and state whether it is invertible.

[3]

Question 9

Use technology to find the determinant and the inverse of the matrix

P = \begin{pmatrix} 2 & -1 & 0 & 3\\ 1 & 0 & 4 & -2\\ 0 & 5 & 1 & 1\\ 3 & 0 & -1 & 2 \end{pmatrix}.

[4]

Question 10

Let

A = \begin{pmatrix} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}.

Use technology to find $A^{-1}$ and compute $\det(A)$ .

[3]

Question 11

Find the inverse of the matrix

A = \begin{pmatrix} 2 & 1 & 0\\ -1 & 1 & 3\\ 0 & 2 & 1 \end{pmatrix}

[6]

Question Type 6: Using technology to find inverse and determinants for nxn matrices where n > 2 Exercises