Cara Mencari Invers Matriks 3×3

>Hello Sohib EditorOnline, in this article we will discuss the method to find the inverse of a 3×3 matrix. This is a crucial concept in linear algebra and finding the inverse of a matrix is useful in solving systems of linear equations, calculating determinants, and more.

What is an inverse matrix?

Before we dive into the methods to find the inverse of a 3×3 matrix, let’s first understand what an inverse matrix is. In general, an inverse matrix is a square matrix, denoted as A^-1, that when multiplied by the original matrix A, results in an identity matrix, denoted as I.

Matrix A	Matrix A-1	Identity Matrix I
a11 a12 a13 a21 a22 a23 a31 a32 a33	b11 b12 b13 b21 b22 b23 b31 b32 b33	1 0 0 0 1 0 0 0 1

When a matrix does not have an inverse, it is referred to as a singular matrix or a non-invertible matrix. In this article, we will focus on finding the inverse of a 3×3 matrix, which may or may not have an inverse.

Method 1: Using the Adjoint Matrix

The first method we will discuss is using the adjoint matrix to find the inverse of a 3×3 matrix. The adjoint matrix, denoted as adj(A), is the transpose of the matrix of cofactors, denoted as C_ij.

Step 1: Find the Matrix of Cofactors

To find the matrix of cofactors, we need to first find the minor of each element in the original matrix. The minor is the determinant of the 2×2 matrix obtained by crossing out the row and column of the element. Then, we need to multiply each minor by either 1 or -1, depending on its position in the matrix, to get the matrix of cofactors.

For example, to find the cofactor C₁₁, we cross out the first row and first column of the matrix, and calculate the determinant of the resulting 2×2 matrix:

a11 a12 a21 a22

= (a11)(a22) – (a12)(a21)

= (1)(4) – (2)(3) = -2

Then, we multiply the cofactor by either 1 or -1, depending on its position in the matrix. In this case, since it is in the first row and first column, we multiply by 1:

C11 = (-1)1+1(-2) = -2

Similarly, we can find the cofactors for the other elements:

C11 = -2	C12 = 1	C13 = -2
C21 = 3	C22 = -5	C23 = 3
C31 = -2	C32 = 1	C33 = -2

These cofactors make up the matrix of cofactors, denoted as C:

C11 C12 C13 C21 C22 C23 C31 C32 C33

= -2 1 -2 3 -5 3 -2 1 -2

Step 2: Find the Adjoint Matrix

Next, we need to find the adjoint matrix, which is the transpose of the matrix of cofactors:

C11	C21	C31
C12	C22	C32
C13	C23	C33

Therefore, the adjoint matrix of matrix A is:

adj(A) = -2 3 -2 1 -5 1 -2 3 -2

Step 3: Find the Inverse Matrix

Finally, we can find the inverse matrix by dividing the adjoint matrix by the determinant of matrix A:

A-1 = adj(A) / det(A)

To find the determinant of matrix A, we can use the rule of Sarrus:

a11 a12 a13 a11 a12 a21 a22 a23 = a21 a22 a31 a32 a33 a31 a32

= (1)(4)(-2) + (2)(3)(3) + (2)(2)(-3) – (2)(4)(3) – (1)(3)(-2) – (2)(2)(1)

= -1

Therefore, the inverse matrix of matrix A is:

A-1 = -2/3 1/3 -2/3 1/3 -5/3 1/3 -2/3 1/3 -2/3

Method 2: Using Elementary Row Operations

The second method we will discuss is using elementary row operations to find the inverse of a 3×3 matrix. Elementary row operations are operations that are performed on the rows of a matrix, and they include:

• Swapping two rows
• Multiplying a row by a nonzero constant
• Adding a multiple of one row to another row

Step 1: Augment the Matrix

First, we need to augment the matrix A with the identity matrix I:

a11 a12 a13 1 0 0 a21 a22 a23 0 1 0 a31 a32 a33 0 0 1

Step 2: Perform Elementary Row Operations

Next, we will perform elementary row operations on the matrix until the left side becomes the identity matrix I:

1 0 0 -2/3 1/3 -2/3

0 1 0 1/3 -5/3 1/3

0 0 1 -2/3 1/3 -2/3

The right-hand side of the augmented matrix is now the inverse matrix of matrix A.

FAQ

1. What is a singular matrix?

A singular matrix, also known as a non-invertible matrix or a degenerate matrix, is a square matrix that does not have an inverse. This can occur when the determinant of the matrix is zero.

2. Can all matrices be inverted?

No, not all matrices can be inverted. A matrix must be a square matrix, and its determinant must not be zero in order for it to be invertible.

3. Why is finding the inverse of a matrix useful?

Finding the inverse of a matrix is useful in solving systems of linear equations, calculating determinants, and more. It can also be used to transform vectors and matrices, and to solve optimization problems.

4. Is there a shortcut to finding the inverse of a matrix?

There is no shortcut to finding the inverse of a matrix, but there are methods that are faster than others, depending on the size of the matrix and the computing resources available.

5. What is the difference between the adjoint matrix and the inverse matrix?

The adjoint matrix is the transpose of the matrix of cofactors, whereas the inverse matrix is the matrix that, when multiplied by the original matrix, results in the identity matrix. The adjoint matrix is used to find the inverse matrix, but they are not the same thing.

Conclusion

In this article, we discussed two methods for finding the inverse of a 3×3 matrix. The first method involved using the adjoint matrix, which is the transpose of the matrix of cofactors, while the second method involved using elementary row operations to augment the matrix with the identity matrix and reduce the left side to the identity matrix. Both methods are valid and useful in different situations, and it’s important to understand both methods in order to be proficient in linear algebra.

Cara Mencari Invers Matriks 3×3