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Inverse of 3×3 Matrix – Formula with Examples

Jo-ann Caballes

Updated on January 10, 2024

Welcome to another exciting adventure in the land of mathematics with Brighterly! Today, we’ll be unraveling the mysteries surrounding the inverse of a 3×3 matrix. It may seem a complex topic at first, but fear not! Our Brighterly team is here to light up your path to understanding.

What is the Inverse of 3×3 Matrix?

In the realm of mathematics, particularly in linear algebra, the concept of a matrix and its inverse is a vital one. For our little math enthusiasts at Brighterly, understanding these mathematical fundamentals can be a piece of cake if we break it down into bite-sized chunks. So, what is the inverse of a 3×3 matrix?

Imagine a square that is divided into nine equal compartments. This square is what mathematicians refer to as a 3×3 matrix. Now, the inverse of this matrix is like a magical mirror that, when multiplied with the original matrix, returns a special matrix called the identity matrix. The identity matrix is like the number 1 of matrices. Just as any number multiplied by 1 remains the same, any matrix multiplied by the identity matrix retains its original values. However, it’s important to note that not all matrices have inverses. Only those matrices which have a non-zero determinant (we will learn about determinants later) have an inverse.

Elements Used to Find Inverse of 3×3 Matrix

Finding the inverse of a 3×3 matrix involves several key elements that need to be computed. Among these elements, the most important are the determinant and the adjoint of the matrix.

Remember our magical mirror analogy? The determinant is like the magical essence that makes the mirror work. If this essence is zero, the mirror doesn’t work, meaning the matrix doesn’t have an inverse. On the other hand, the adjoint of a matrix is like the reflection that our magical mirror (the inverse) shows when a matrix looks into it.

Both the determinant and adjoint are computed using specific methods and have a special role in the formula for finding the inverse of a 3×3 matrix. So, let’s delve deeper and understand the process of finding the determinant and adjoint of a 3×3 matrix.

Adjoint of a 3×3 Matrix

The adjoint of a 3×3 matrix is obtained by a multi-step process. First, we compute the cofactor matrix. This involves taking each element of the matrix, leaving out its corresponding row and column, and finding the determinant of the remaining 2×2 matrix (also called the minor). The sign of the cofactor is determined by the position of the element in the matrix.

Next, we create a cofactor matrix by replacing each element with its corresponding cofactor. Finally, we take the transpose of the cofactor matrix, i.e., we interchange the rows and columns, to get the adjoint. While it sounds complicated, with practice and patience, computing the adjoint becomes a fun puzzle to solve.

Determinant of a 3×3 Matrix

The determinant of a 3×3 matrix is a value that gives us information about the matrix and plays an essential role in finding the inverse. We calculate the determinant by summing up the products of diagonals in one direction and subtracting the sum of products of diagonals in the other direction.

We should remember that if the determinant is zero, the matrix does not have an inverse. This is because we divide by the determinant in the inverse formula, and we all know that dividing by zero is a big no-no in mathematics!

How to Find the Inverse of 3 x 3 Matrix?

Now that we understand the ingredients, let’s look at how to find the inverse of a 3×3 matrix.

Once we’ve computed the determinant and the adjoint, the inverse can be found by dividing the adjoint by the determinant. More precisely, each element of the adjoint is divided by the determinant to get the corresponding element of the inverse matrix. It’s like sharing pieces of a pie equally among friends.

The process might sound complicated at first, but with our step-by-step guide, you’ll be able to find the inverse of a 3×3 matrix in no time.

Inverse of 3×3 Matrix Formula

So, what is the inverse of 3×3 matrix formula? The formula for the inverse of a 3×3 matrix is given by Inverse of A = 1/Determinant(A) * Adjoint(A). Here, A represents the matrix, the Determinant(A) is the determinant of the matrix, and the Adjoint(A) is the adjoint of the matrix.

This formula is like the secret recipe that helps us find the magical mirror (inverse) for any 3×3 matrix.

Finding Inverse of 3×3 Matrix Using Row Operations

There’s another technique called row operations that can be used for finding the inverse of a 3×3 matrix. In this method, we perform the same operation (like adding, subtracting, or multiplying rows) on the given matrix and an identity matrix side by side. The aim is to transform the given matrix into the identity matrix. The transformed version of the identity matrix will be the inverse of the original matrix.

It’s a bit like a game of matching shadows, where we manipulate the given matrix until it matches the shadow (identity matrix) and find the inverse matrix in the process.

Solving System of 3×3 Equations Using Inverse

Did you know? We can use the inverse of a matrix to solve systems of 3×3 equations. Here, we express the system of equations in matrix form. Then, we find the inverse of the coefficient matrix and multiply it with the matrix of constants to get the solution matrix. It’s a handy tool when dealing with multiple equations with multiple variables.

This is where the power of the inverse matrix shines, transforming complex equation solving into a simple multiplication process.

Inverse of 3×3 Matrix Examples

Let’s explore some inverse of 3×3 matrix examples to understand these concepts better.

Suppose we have a matrix A, and we want to find its inverse. We first calculate the determinant and the adjoint using the methods mentioned above. Then, we apply the formula to find the inverse. With clear examples and step-by-step instructions, you’ll be mastering these concepts in no time.

Practice Questions on Inverse of 3×3 Matrix

At Brighterly, we truly believe in the power of practice. So, let’s get your math neurons firing with these practical questions! They’re designed to cement your understanding of how to find the inverse of a 3×3 matrix.

1. Question 1:

   Find the inverse of the following 3×3 matrix, if it exists.

   A = [ [1, 2, 3], [0, 1, 4], [5, 6, 0] ]

2. Question 2:

   Determine whether the following 3×3 matrix has an inverse. If yes, compute the inverse.

   B = [ [7, 2, 1], [0, 3, -1], [-3, 4, 1] ]

3. Question 3:

   Compute the inverse of the given 3×3 matrix, if it exists.

   C = [ [-2, -3, 1], [1, 0, 4], [5, -1, 2] ]

4. Question 4:

   Check if the following 3×3 matrix has an inverse. If it does, find the inverse.

   D = [ [1, 0, 0], [0, -1, 0], [0, 0, 1] ]

5. Question 5:

   Calculate the inverse of the following 3×3 matrix, if possible.

   E = [ [3, 2, -1], [1, 1, 0], [0, 1, 1] ]

Remember, there’s no need to rush. Take your time, use the step-by-step guide we’ve provided above, and soon, you’ll be finding the inverse of a 3×3 matrix like a pro! Happy solving with Brighterly!

Conclusion

Congratulations! You’ve journeyed with us through the fascinating world of matrices and discovered the secrets of the inverse of a 3×3 matrix. At Brighterly, we believe in making learning engaging, fun, and most importantly, understandable. We’ve learned how a matrix can have a magical twin, how this twin is found, and even its uses in solving complex systems of equations.

Remember, the key to unlocking your mathematical prowess lies in practice. So, dive into our curated set of practice questions, and don’t hesitate to refer back to this guide as many times as you need. Together with Brighterly, you can brighten your mathematical journey, one concept at a time.

Frequently Asked Questions on Inverse of a 3×3 Matrix

Does every matrix have an inverse?

Not every matrix has an inverse. In fact, a matrix will only have an inverse if its determinant is non-zero. If the determinant is zero, the matrix is known as a singular matrix, and it does not have an inverse.

What happens when a matrix is multiplied by its inverse?

When a matrix is multiplied by its inverse, the result is a special matrix called the identity matrix. This matrix has ones along the diagonal (from the top left to the bottom right) and zeros everywhere else. It’s the equivalent of the number 1 in the world of matrices!

What is the role of the determinant in finding the inverse?

The determinant plays a crucial role in finding the inverse of a matrix. The inverse of a matrix is found by dividing the adjoint of the matrix by the determinant. If the determinant is zero, this division isn’t possible, hence the matrix doesn’t have an inverse.

How does the inverse of a matrix help in solving systems of equations?

The inverse of a matrix is a powerful tool when it comes to solving systems of equations. If we can express the system of equations in matrix form, we can find the inverse of the coefficient matrix and multiply it with the matrix of constants. The resulting matrix will be the solution to the system of equations. This method transforms the complex process of solving multiple equations into a simpler process of matrix multiplication.

Jo-ann Caballes

5 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.