Q&A for How to Find the Inverse of a 3x3 Matrix

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How do you use elementary row operations to get the inverse of a matrix?

Mario Banuelos, PhD
Associate Professor of Mathematics
Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

Mario Banuelos, PhD

Associate Professor of Mathematics

Expert Answer

Begin by setting up the system [A | I] where I is the identity matrix. Then, use elementary row operations to make the left hand side of the system reduce to I. The resulting system will be [I | A⁻¹] where A⁻¹ is the inverse of A.

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How do I evaluate the inverse of the matrix {1 2 -4}{0 -2 3}{5 0 4}?

Community Answer

Find the determinant, then determine the co-factor matrix. Find the adj of the co-factor matrix, then divide through each term by the determinant.

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Are there any shortcuts for finding the inverse of a 3x3 matrix?

Community Answer

The methods shown in the article is as simple as it gets unfortunately; you can do drills and make up your own 3x3 matrices to find the inverse of in order to remember the steps.

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How can I create a 3x3 matrix without any fractions in its original form and inverse form?

Community Answer

Create a 3 x 3 matrix whose determinant is 1 and whose elements are all integers. The associated inverse matrix will have only integer elements as well.

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Can you please help me find the answer to this problem? "Inverse of matrix 3x3|(1&1&0@1&1&1@0&2&1)|"

Community Answer

Just follow the steps; your determinant should be -2, and your matrix of co-factors should be (-1&1&1@1&1&1@2&2&0). From there, apply the +- matrix and then divide by the determinant.

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What is a matrix?

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A matrix is a generalization of a vector.

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Can I multiply a row in a matrix by -1?

Community Answer

Yes, you can multiply a row in a matrix by -1 as long as you multiply all numbers in a row.

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How would I know if the inverse of a matrix does not exist?

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You would transform your matrix into row-echelon form. Once you do, you can see that if the matrix is a perfect identity matrix, then the inverse exists. Otherwise, it doesn't.

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Is it necessary to A = IA for elementary row operation, or can it be written as A = AI?

Community Answer

A = AI is written for elementary column operation, but elementary row operation is always written A = IA.

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