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Inverse Matrix
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In these lessons, we will learn about the identity matrix and inverse matrices.
We also feature a matrix calculator that will help you to find the inverse of a 3×3 matrix.
Use it to check your answers.
The identity matrix and the inverse of a matrix are closely connected.
The Key Relationship:
The inverse of a matrix “undoes” the effect of the original matrix in matrix multiplication. The identity matrix is the result of this “undoing,” similar to how multiplying a number by its reciprocal (e.g., 5 × 1/5 = 1) gives you 1.
Analogy to Numbers:
Think of it like this:
In regular multiplication, the number 1 is the multiplicative identity because any number multiplied by 1 equals itself (e.g., 7 × 1 = 7).
The reciprocal of a number (its inverse) is what you multiply it by to get 1 (e.g., 7 × 1/7 = 1).
In the same way:
The identity matrix I is the multiplicative identity for matrices.
When you multiply a matrix A with its inverse you will get the identity matrix I.
Learn how to calculate the inverse of a matrix
A square matrix, I is an identity matrix if the product of I and any square
matrix A is A.
i.e. IA = AI = A
For a 2 × 2 matrix, the identity matrix for multiplication is
When we multiply a matrix with the identity matrix, the original matrix is unchanged.
If the product of two square matrices, P and Q, is the identity matrix
then Q is an inverse matrix of P and P is the inverse matrix of Q.
i.e. PQ = QP = I
The inverse matrix of A is denoted by A-1. (read as “A inverse”)
AA-1 = A-1A = I
Note that the inverse of A-1 is A.
Example:
Given that B is the inverse of A, find the values of x and y.
Solution:
AB =
Since B is an inverse of A, we know that AB = I
1 – 2y = 1
2y = 0
y = 0
2x = 1
x =
The Identity Matrix
This video introduces the identity matrix and illustrates the properties of the identity matrix.
A n × n square matrix with a main diagonal of 1’s and all other elements 0’s is called the identity matrix In.
If A is a m × n matrix, then ImA = A and AIn = A.
Is A is a n × n square matrix, then
AIn = InA = A.
Determining a 2×2 Inverse Matrix Using a Formula
This video explains the formula used to determine the inverse of a 2x2 matrix, if one exists.
Example:
Determine the inverse of matrix A.
How to find the inverse of a 2×2 matrix using the inverse formula?
Identity matrix and inverse matrix
This matrix calculator will help you find the inverse of a 3×3 matrix.
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