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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

Skill question

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

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

Question 2

Skill question

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.

Question 3

Skill question

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.

Question 4

Skill question

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}

Question 5

Skill question

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

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

Question 6

Skill question

Let

A = \begin{pmatrix} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}. $$Use technology to find $A^{-1}$ and compute $\det(A)$.

Question 7

Skill question

Compute the determinant of the matrix with fractional entries:

Q = \begin{pmatrix} \tfrac12 & 2 & 0\\ -1 & \tfrac34 & 1\\ 2 & -1 & \tfrac13 \end{pmatrix}.

Question 8

Skill question

Find the inverse of the matrix

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

Question 9

Skill question

Compute the determinant of the following $4\times4$ matrix:

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

Question 10

Skill question

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}.

Question 11

Skill question

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}.

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