Determine whether an inverse exists and calculate it practice questions | General Mathematics (Year 12, VCE Units 3 & 4, Matrices, Linear operations on matrices)

Determinants and inverse existence

For a square matrix, calculate the determinant of the whole matrix before deciding whether an inverse exists. A non-zero determinant permits an inverse, which can be checked by multiplying the matrix and its proposed inverse

\det(A)\ne0\Longleftrightarrow A^{-1}\text{ exists}

Use the whole matrix For an order greater than 2, include every row and column in the determinant calculation \(\displaystyle D=\begin{bmatrix}2&1&0\\1&2&1\\0&1&2\end{bmatrix},\quad\det(D)=4\)
Use the 2 × 2 rule at order 2 For a 2 × 2 matrix, subtract the product of the other diagonal from the leading-diagonal product \(\displaystyle \det\begin{bmatrix}4&1\\2&3\end{bmatrix}=4\times3-1\times2=10\)
Test for zero A zero determinant means that the square matrix has no inverse \(\displaystyle \det(A)=0\Rightarrow A^{-1}\text{ does not exist}\)
Check the proposed inverse A correct inverse gives the identity matrix when multiplied in either order \(\displaystyle AA^{-1}=I_n=A^{-1}A\)

Worked example

Find the inverse of the matrix B and verify the result by multiplication

B=\begin{bmatrix}1&2&0\\0&1&3\\2&0&1\end{bmatrix}

\(\displaystyle \det(B)=1(1\times1-3\times0)-2(0\times1-3\times2)=13\)Expand along the first row
\(\displaystyle 13\ne0\Rightarrow B^{-1}\text{ exists}\)Because the determinant is non-zero, B has an inverse
\(\displaystyle C_{12}=-(0\times1-3\times2)=6,\quad C=\begin{bmatrix}1&6&-2\\-2&1&4\\6&-3&1\end{bmatrix}\)Calculate the signed 2 by 2 minors to form the cofactor matrix
\(\displaystyle \operatorname{adj}(B)=C^{\mathsf T}=\begin{bmatrix}1&-2&6\\6&1&-3\\-2&4&1\end{bmatrix}\)Transpose the cofactor matrix
\(\displaystyle B^{-1}=\frac{1}{13}\begin{bmatrix}1&-2&6\\6&1&-3\\-2&4&1\end{bmatrix}\)Divide the adjugate by the determinant
\(\displaystyle BB^{-1}=\frac{1}{13}\operatorname{diag}(13,13,13)=I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)The product is the identity matrix, so the proposed inverse is verified

Common mistakes

Using only a leading 2 × 2 block

✕\(\displaystyle \det(B)=1\times1-2\times0=1\)✓\(\displaystyle B=\begin{bmatrix}1&2&0\\0&1&3\\2&0&1\end{bmatrix},\quad\det(B)=13\)

A determinant for a matrix larger than 2 × 2 must use the whole matrix, not just its leading block

Assuming every square matrix is invertible

✕\(\displaystyle B\text{ square}\Rightarrow B^{-1}\text{ exists}\)✓\(\displaystyle \det(B)\ne0\Rightarrow B^{-1}\text{ exists}\)

Being square does not rule out a zero determinant, so inverse existence must be tested

Taking entrywise reciprocals

✕\(\displaystyle C=\begin{bmatrix}5&1\\2&2\end{bmatrix}\Rightarrow C^{-1}=\begin{bmatrix}\frac{1}{5}&1\\\frac{1}{2}&\frac{1}{2}\end{bmatrix}\)✓\(\displaystyle C^{-1}=\frac{1}{8}\begin{bmatrix}2&-1\\-2&5\end{bmatrix}\)

A matrix inverse is not formed by taking the reciprocal of each entry; it must satisfy multiplication to the identity

Difficulty levels

introCalculate a 2 × 2 determinant and decide whether an inverse can exist

standardCalculate an exact 2 × 2 inverse when the determinant is non-zero

challengeWork with a larger square matrix and verify an exact inverse by multiplication

examDiagnose determinant and inverse errors in a higher-order matrix problem

Questions about this topic

Does every square matrix have an inverse?+

No. A square matrix has an inverse only when its determinant is non-zero

Why must the determinant of a larger matrix use every entry?+

The determinant is a property of the whole square matrix, so ignoring rows or columns can give a value for a different matrix

How is an inverse verified?+

Multiply the matrix by the proposed inverse. The result must be the identity matrix of the matching order

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 10: Matrices

In this chapter

State the order of a matrix and locate its elements Identify row, column, square, symmetric and zero matrices Read information from a labelled data matrix Read, transpose and classify a matrix Construct a matrix from labelled information or an entry rule 10A Matrix notation, types and construction: Solve a matrix equation 10A Matrix notation, types and construction: Construct, apply and invert a permutation matrix Add and subtract matrices Find unknown entries in a matrix equation Multiply a matrix by a scalar Combine scalar multiples of matrices 10B Linear operations on matrices: Determine whether an inverse exists and calculate it Decide whether a matrix product is defined and give its order Multiply matrices Evaluate powers of a square matrix Identify and construct the identity matrix 10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrix Find the determinant of a 2 × 2 matrix Find the inverse of a 2 × 2 matrix Write simultaneous equations as a matrix equation Solve simultaneous equations using the inverse matrix 10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it 10D Determinants, inverses and matrix equations: Solve a matrix equation Use matrix multiplication to answer an application question Use row and column matrices of ones to extract totals Represent a network in a matrix Identify the network a connection matrix describes Interpret connection sums and symmetry Count routes using powers of a connection matrix Construct and analyse relation matrices