Multiplicative Identity Matrix
Understand the multiplicative identity matrix in Grade 10 algebra. Identify the square matrix I with diagonal 1s and off-diagonal 0s that satisfies A×I=I×A=A for any matrix A.
Key Concepts
The multiplicative identity matrix, denoted as $I$, is a square matrix where all elements on the main diagonal are ones and all other elements are zeros. The product of any matrix $A$ and the appropriate identity matrix $I$ is always $A$. $$AI = IA = A$$.
For $A = \begin{bmatrix} 5 & 1 \\ 2 & 8 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then $AI = \begin{bmatrix} 5 & 1 \\ 2 & 8 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 1 \\ 2 & 8 \end{bmatrix}$. For $B = \begin{bmatrix} 4 & 1 & 7 \\ 2 & 0 & 5 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $BI = \begin{bmatrix} 4 & 1 & 7 \\ 2 & 0 & 5 \end{bmatrix}$.
Meet the superhero of matrices: the Identity Matrix, or $I$! It's the matrix version of the number 1. Multiplying any matrix by $I$ is like giving it a high five—it leaves the original matrix completely unchanged. This special square matrix has a slick diagonal of ones and is your go to for preserving a matrix's identity during calculations.
Common Questions
What is the multiplicative identity matrix?
The identity matrix I is a square matrix with 1s on the main diagonal and 0s everywhere else. Multiplying any matrix A by I gives A: A × I = I × A = A.
What does the 2×2 identity matrix look like?
The 2×2 identity matrix is [[1,0],[0,1]]. For any 2×2 matrix A, multiplying by this identity matrix returns A unchanged in both row-left and column-right multiplication.
Why is the identity matrix important in matrix algebra?
The identity matrix is the matrix equivalent of the number 1. It is used in finding matrix inverses: A × A⁻¹ = I. It is also fundamental in solving matrix equations.