Identity Matrix

High School

Definition

Sometimes known as a unit matrix. The n x n identity matrix has n rows and n columns. The upper left and bottom right diagonal consists of 1s, and the other entries in the matrix are all 0s. By multiplying another matrix by an identity matrix, it'll leave the original matrix the same.

Worked examples

\(I_2 = \begin{bmatrix} 1 & 0 \)
\( 0 & 1 \end{bmatrix}, \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \)
\( 0 & 1 & 0 \)
\( 0 & 0 & 1 \end{bmatrix}\)

The 2×2 and 3×3 identity matrices have 1s on the main diagonal and 0s everywhere else.

\(\begin{bmatrix} 1 & 0 \)
\( 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 5 \)
\( 7 & 2 \end{bmatrix} = \begin{bmatrix} 3 & 5 \)
\( 7 & 2 \end{bmatrix}\)

Multiplying by the identity matrix leaves the original matrix unchanged.

Common mistakes

\(I_2 = \begin{bmatrix} 0 & 0 \)
\( 0 & 0 \end{bmatrix}\) → \(I_2 = \begin{bmatrix} 1 & 0 \)
\( 0 & 1 \end{bmatrix}\) The identity has 1s on the diagonal, not 0s; the zero matrix is different.
\(I_2 = \begin{bmatrix} 1 & 1 \)
\( 1 & 1 \end{bmatrix}\) → \(I_2 = \begin{bmatrix} 1 & 0 \)
\( 0 & 1 \end{bmatrix}\) Only the main diagonal entries are 1; all off-diagonal entries must be 0.
The identity matrix works only on square matrices. → \(I_m A_{m \times n} = A\) and \(A_{m \times n} I_n = A\) You can multiply non-square matrices by identity matrices of compatible dimensions.

Where you'll use it next

Identity matrices are central to solving systems of equations using matrix inverses, finding eigenvalues and eigenvectors, and understanding linear transformations in higher algebra and linear algebra courses.

Found in 1 StudyPug lesson

Mastering Identity Matrices in Linear Algebra

Grade 12 Math

Dive into the world of identity matrices! Discover their unique properties, learn how to construct them, and explore their crucial role in linear algebra and real-world applications.

12th Grade12th