# 11.3: Identity Matrix

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Read sections Sections 6.2 and 6.3 of the Stephen Boyd and Lieven Vandenberghe Applied Linear algebra book covers more about matrixes.

An identity matrix is a special square matrix (i.e. $$n=m$$) that has ones in the diagonal and zeros other places. For example the following is a $$3×3$$ identity matrix:

$\begin{split} I_3 = \left[ \begin{matrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \end{split} \nonumber$

We always denote the identity matrix with a capital $$I$$. Often a subscript is used to denote the value of $$n$$. The notations $$I_{n \times n}$$ and $$I_{n}$$ are both acceptable.

An identity matrix is similar to the number 1 for scalar values. I.e. multiplying a square matrix $$A_{n \times n}$$ by its corresponding identity matrix $$I_{n \times n}$$ results in itself $$A_{n \times n}$$.

##### Do This

Pick a random $$3 \times 3$$ matrix and multiply it by the $$3 \times 3$$ Identity matrix and show you get the same answer.

##### Question

Consider two square matrices $$A$$ and $$B$$ of size $$n \times n$$. $$AB=BA$$ is NOT true for many $$A$$ and $$B$$. Describe an example where $$AB=BA$$ is true? Explain why the equality works for your example.

##### Question

The following matrix is symmetric. What are the values for $$a$$, $$b$$, and $$c$$? (HINT you may want to look online or in the Boyd book for a definition of matrix symmetry)

$\begin{split} \left[ \begin{matrix} 3 & 5 & a\\ b & 8 & 4 \\ -3 & c & 3 \end{matrix} \right] \end{split} \nonumber$

This page titled 11.3: Identity Matrix is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.