# How to Find the Inverse of a 3×3 . Matrix

Invertible matrix
Invertible matrix

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Matrix inverses are an essential concept in linear algebra, particularly when solving systems of linear equations. They provide a way to “reverse” the transformation represented by a matrix, allowing us to find the original values or variables. While finding the inverse of a 2×2 matrix is well-known and relatively straightforward, the process becomes more complex when dealing with larger matrices. In this article, we will explore the steps involved in finding the inverse of a 3×3 matrix. By understanding and applying these techniques, we can effectively manipulate matrices to solve a variety of mathematical problems.

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The inverse is often used in calculus to simplify difficult problems in other ways. For example, it is easier to multiply by the reciprocal of a fraction than to divide by the number directly. This is the inverse. Similarly, since there is no fractional sign for the matrix, you will have to multiply by its inverse. Manually calculating the inverse of a 3×3 matrix can be tedious, but it’s a problem worth considering. You can also use an advanced graphing calculator to do this.

## Steps

### Create an additional matrix to find the inverse

• To find the inverse of a 3×3 matrix, you must first calculate its determinant.
• To review how to find the determinant of a matrix, refer to the article Finding the determinant of a 3×3 matrix.

• Another way of understanding transposition is that you would rewrite the matrix so that the first row becomes the first column, the middle row becomes the middle column, and the third row becomes the third column. Notice the color factor in the illustration above and notice the new placement of the numbers.

• In the above example, if you want to find the submatrix of the element located in row two, column one, you highlight five parts from the second row and first column. The remaining four elements are the corresponding submatrix.
• Find the determinant of each submatrix by multiplying diagonally and subtracting the two products, as shown in the figure above.

• When specifying the sign, the sign of the first element of the leading row will be preserved. The sign of the second element is inverted. The sign of the third element is preserved. Continue in the same way for the rest of the matrix. Note that the (+) or (-) signs in the reference chart do not indicate whether the element will have a positive or negative sign at the end. They just indicate that the elements will stay the same (+) or change the sign (-).
• Refer to matrix basics for more on algebraic subsection.
• The final result that we get in this step is the addition matrix of the original matrix. It is sometimes called the conjugate matrix and is denoted by Adj(M).

• The sample matrix presented in the illustration has a determinant of 1. Thus, dividing every element of the matrix by the determinant, we get itself (you won’t always be so lucky) .
• Instead of division, some literature presents this step as multiplying every element of M by 1/det(M). Mathematically, they are equivalent.

### Linear row reduction to find the inverse matrix

• Remember that the unit matrix is a special matrix where every element on the main diagonal, running from the upper left corner to the lower right corner, is equal to 1 and all elements in the remaining positions are equal to 0.

• Remember that row reduction is performed as a combination of scalar multiplication and row addition or subtraction, to isolate the individual elements of the matrix.

### Find the inverse matrix using a pocket calculator

• If you want to enter a negative number, use the negative (-) button of the calculator, not the minus key. The matrix function will not read properly.
• If needed, you can use the arrow keys on your calculator to move through the matrix.

• Don’t use the ^ button on a calculator when trying to enter A^-1 with individual presses. The computer will not understand this math.
• If you get an error message when you press the inverse key, chances are your original matrix is not invertible. Maybe you should go back and compute the determinant to determine if that’s the reason for the error.

• Maybe your calculator has a function to automatically convert decimals to fractions. For example, when using the TI-86, you can go into the Math function, select Misc then Frac and press Enter. Decimals are automatically represented as fractions.

• You can follow these steps to find the inverse of a matrix that contains not only numbers but also variables, unknowns, or even algebraic expressions.
• Write down all the steps because finding the inverse of a 3×3 matrix just by doing math is extremely difficult.
• There are computer programs that help you find the inverse matrix [18] XResearch Source , up to and including 30×30 matrices.
• Whatever method is used, check the accuracy of the result by multiplying M by M -1 . You will find that M*M -1 = M -1 *M = I. Where, I is the unit matrix, made up of 1’s along the main diagonal and 0’s on the main diagonal. other locations. If you don’t get such results, you must have made a mistake somewhere.

## Warning

• Not every 3×3 matrix has an inverse. If the determinant is 0, the matrix is not invertible (Note that in the formula we divide by det(M).Dividing by zero is an undefined operation).

The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.

The inverse is often used in calculus to simplify difficult problems in other ways. For example, it is easier to multiply by the reciprocal of a fraction than to divide by the number directly. This is the inverse. Similarly, since there is no fractional sign for the matrix, you will have to multiply by its inverse. Manually calculating the inverse of a 3×3 matrix can be tedious, but it’s a problem worth considering. You can also use an advanced graphing calculator to do this.

In conclusion, finding the inverse of a 3×3 matrix involves a step-by-step process that can be accomplished using various techniques such as the adjugate method or the row reduction method. The key factor in finding the inverse is ensuring that the matrix is non-singular, meaning that its determinant is not zero. Once the inverse is obtained, it can be used to solve systems of linear equations, perform transformations, or simplify complex calculations. Understanding how to find the inverse of a 3×3 matrix is a valuable skill in mathematics, engineering, and other fields that involve linear algebra.

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