Finding the inverse of a square matrix using an augmented matrix is a fundamental method that leverages elementary row operations to transform the original matrix into the identity matrix, simultaneously revealing its inverse.
Understanding the Augmented Matrix Method
To calculate the inverse of a matrix, say matrix A, using elementary row transformations, we first construct an augmented matrix. This augmented matrix combines matrix A with an identity matrix (I) of the same order, written as [A | I]. The core idea is to apply a series of row operations to the entire augmented matrix until the left side (matrix A) is transformed into the identity matrix (I). As this transformation occurs, the right side (initially I) will simultaneously transform into the inverse of A, denoted as A⁻¹. The final form of the augmented matrix will then be [I | A⁻¹].
This method is applicable only to square matrices that are invertible (i.e., non-singular, meaning their determinant is non-zero).
Step-by-Step Process to Find the Inverse
Follow these steps to find the inverse of a matrix A using the augmented matrix method:
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Form the Augmented Matrix:
- Begin by creating an augmented matrix by placing the given square matrix A on the left side and an identity matrix I of the same dimensions on the right side.
- This is represented as
[A | I].
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Apply Elementary Row Operations:
- The goal is to transform the left side (matrix A) into the identity matrix (I) by applying a sequence of elementary row operations. These operations include:
- Swapping two rows: Interchanging the positions of any two rows.
- Multiplying a row by a non-zero scalar: Multiplying all elements in a row by a constant number (except zero).
- Adding a multiple of one row to another row: Replacing a row with the sum of itself and a multiple of another row.
- It is crucial to apply each operation to the entire augmented matrix (both sides simultaneously).
- The goal is to transform the left side (matrix A) into the identity matrix (I) by applying a sequence of elementary row operations. These operations include:
-
Transform A into I:
- Work systematically, usually column by column, to achieve the identity matrix form on the left.
- A common strategy is to first create zeros below the main diagonal, then create ones on the main diagonal, and finally create zeros above the main diagonal. This process is known as Gauss-Jordan elimination.
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Identify the Inverse:
- Once the left side of the augmented matrix has been transformed into the identity matrix
I, the matrix on the right side will be the inverse of the original matrix A (A⁻¹). - The augmented matrix will now be in the form
[I | A⁻¹].
- Once the left side of the augmented matrix has been transformed into the identity matrix
Example: Finding the Inverse of a 2x2 Matrix
Let's find the inverse of the matrix $A = \begin{pmatrix} 2 & 1 \ 4 & 3 \end{pmatrix}$ using the augmented matrix method.
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Form the augmented matrix:
$$ [A | I] = \begin{pmatrix} 2 & 1 & | & 1 & 0 \ 4 & 3 & | & 0 & 1 \end{pmatrix} $$ -
Apply row operations:
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Step 1: Make the leading element of R1 equal to 1.
Apply $R1 \to \frac{1}{2}R1$:
$$ \begin{pmatrix} 1 & 1/2 & | & 1/2 & 0 \ 4 & 3 & | & 0 & 1 \end{pmatrix} $$ -
Step 2: Make the element below the leading 1 in R1 equal to 0.
Apply $R2 \to R2 - 4R1$:
$$ \begin{pmatrix} 1 & 1/2 & | & 1/2 & 0 \ 0 & 1 & | & -2 & 1 \end{pmatrix} $$
(Calculation for R2: $4 - 4(1) = 0$, $3 - 4(1/2) = 3 - 2 = 1$, $0 - 4(1/2) = -2$, $1 - 4(0) = 1$) -
Step 3: Make the element above the leading 1 in R2 equal to 0.
Apply $R1 \to R1 - \frac{1}{2}R2$:
$$ \begin{pmatrix} 1 & 0 & | & 3/2 & -1/2 \ 0 & 1 & | & -2 & 1 \end{pmatrix} $$
(Calculation for R1: $1 - \frac{1}{2}(0) = 1$, $1/2 - \frac{1}{2}(1) = 0$, $1/2 - \frac{1}{2}(-2) = 1/2 + 1 = 3/2$, $0 - \frac{1}{2}(1) = -1/2$)
-
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Identify the inverse:
The left side is now the identity matrix $I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$.
Therefore, the inverse of matrix A is $A^{-1} = \begin{pmatrix} 3/2 & -1/2 \ -2 & 1 \end{pmatrix}$.
Important Considerations
- Non-invertible Matrices: If, during the row operations, you end up with a row of all zeros on the left side of the augmented matrix, it means the original matrix is singular (its determinant is zero) and thus does not have an inverse.
- Computational Intensity: While effective for small matrices, this method can become computationally intensive and prone to error for larger matrices. For very large matrices, numerical methods or specialized software are typically used.
- Uniqueness: If an inverse exists, it is unique.
