The identity matrix
An identity matrix is square and has 1s on its leading diagonal, with 0s elsewhere. Match the identity order to the number of rows on the left or the number of columns on the right
\[\operatorname{order}(A)=m\times n,\quad I_mA=A=AI_n\]
- Use a square order An identity matrix of order n has n rows and n columns \(\displaystyle I_3\text{ has order }3\times3\)
- Put 1s on the leading diagonal The entries from the top left to the bottom right are all 1. \(\displaystyle I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
- Put 0s off the diagonal Every position not on the leading diagonal contains 0. \(\displaystyle \left(I_3\right)_{12}=0\)
- Keep a matrix unchanged For a matrix with 2 rows and 3 columns, use I_2 on the left and I_3 on the right \(\displaystyle \operatorname{order}(A)=2\times3,\quad I_2A=A=AI_3\)
Worked example
Construct the identity matrix of order 3.
\[I_3\]
- \(\displaystyle \operatorname{order}(I_3)=3\times3\)An identity matrix of order 3 has 3 rows and 3 columns
- \(\displaystyle (1,1),(2,2),(3,3)\text{ contain }1\)Place 1 in each position on the leading diagonal
- \(\displaystyle 3\times3-3=6\text{ off-diagonal positions contain }0\)All six positions outside the leading diagonal contain 0.
- \(\displaystyle I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)The completed square matrix has 1s on the leading diagonal and 0s elsewhere
Common mistakes
Filling every position with 1
✕\(\displaystyle I_3=\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}\)✓\(\displaystyle I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
The entries away from the leading diagonal are 0, not 1.Reversing the diagonal values
✕\(\displaystyle \begin{bmatrix}0&1&1\\1&0&1\\1&1&0\end{bmatrix}\)✓\(\displaystyle \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
The leading diagonal must contain 1s and the off-diagonal positions must contain 0sUsing the wrong order
✕\(\displaystyle I_3=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)✓\(\displaystyle I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
The subscript gives the number of rows and columns, so I_3 must be a 3 × 3 matrixDifficulty levels
introConstruct or recognise a 2 × 2 identity matrix
standardConstruct or recognise a 3 × 3 identity matrix and its zero entries
challengeUse the identity property for a matrix of a stated order
examSelect the identity matrix or the product that uses its defining property
Questions about this topic
Why must an identity matrix be square?+
An identity has one leading-diagonal 1 in every row and column, with 0s elsewhere. An identity matrix of order n leaves a matrix unchanged when used on the left of a matrix with n rows or on the right of a matrix with n columns
Where are the 1s placed?+
The 1s are placed from the top-left corner to the bottom-right corner on the leading diagonal
What is the difference between an identity matrix and a zero matrix?+
A zero matrix has 0 in every position, while an identity matrix has 1s on the leading diagonal and 0s elsewhere
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals