Inverse of 3x3 Matrix

Before going to see how to find the inverse of a 3x3 matrix, let us recall the what the inverse mean. The inverse of a number is a number which when multiplied by the given number results in the multiplicative identity, 1. In the same way, the product of a matrix A and its inverse A^-1 gives the identity matrix, I. i.e., AA^-1 = A^-1A = I. Let us see how to find the inverse of 3x3 matrix.

Let us see the formula for finding the inverse of 3x3 matrix along with some other ways of finding it. Also, we will see some examples of finding the inverse of a 3x3 matrix.

The inverse of a 3x3 matrix, say A, is a matrix of the same order denoted by A^-1 where AA^-1 = A^-1A = I, where I is the identity matrix of order 3x3. i.e., I = \(\left[\begin{array}{rr}1 & 0 & 0 \\ 0&1&0 \\ 0 & 1&0 \end{array}\right]\). For example, if A = \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\) then A^-1 = \(\left[\begin{array}{rr}3 / 16 & 1 / 4 & -5 / 16 \\
1 / 4 & 0 & 1 / 4 \\
-5 / 16 & 1 / 4 & 3 / 16 \\ \end{array}\right]\). One can easily multiply these matrices and verify whether AA^-1 = A^-1A = I. We will see how to find the inverse of a 3x3 matrix in the upcoming section.

Before going to know how to find the inverse of 3x3 matrix, let us see how to find the determinant and adjoint of 3x3 matrix. Let us use this same example (as in the previous section) in each explanation.

Adjoint of a 3x3 Matrix

The adjoint of a matrix A is obtained by finding the transpose of the cofactor matrix of A. To know how to find the adjoint of a matrix in detail click here. The cofactor of any element of a 3x3 matrix is the determinant of 2x2 matrix that is obtained by removing the row and the column containing the element. Also, we write alternate + and - signs while finding the cofactors. Here is an example.

Let A = \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\).

Then its cofactor matrix is:

\(\left[\begin{array}{rr}\left|\begin{array}{ll} 1 & 2 \\
2 & 1
\end{array}\right| & -\left|\begin{array}{cc}
2 & 2 \\
-1 & 1
\end{array}\right| & -\left|\begin{array}{cc}
2 & 1 \\
-1 & 2
\end{array}\right|\\
-\left|\begin{array}{cc}
2 & -1 \\
2 & 1
\end{array}\right| & \left|\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array}\right|&-\left|\begin{array}{cc}
1 & 2 \\
-1 & 2
\end{array}\right| \\ \left|\begin{array}{cc}
2 & -1 \\
1 & 2
\end{array}\right|& -\left|\begin{array}{rr}
1 & -1 \\
2 & 2
\end{array}\right|&\left|\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right| \end{array}\right]\)

Each 2x2 determinant is obtained by multiplying diagonals and subtracting the products (from left to right).

So the cofactor matrix = \(\left[\begin{array}{ccc}
1-4 & -(2+2) & 4+1 \\
-(2+2) & 1-1 & -(2+2) \\
4+1 & -(2+2) & 1-4
\end{array}\right]\)

= \(\left[\begin{array}{rrr}
-3 & -4 & 5 \\
-4 & 0 & -4 \\
5 & -4 & -3
\end{array}\right]\)

By transposing the cofactor matrix, we get the adjoint matrix.

So adj A = \(\left[\begin{array}{ccc}
-3 & -4 & 5 \\
-4 & 0 & -4 \\
5 & -4 & -3
\end{array}\right]\).

(Of course, we have got both the cofactor matrix and adjoint matrix to be the same in this case. But it may not happen always).

Determinant of a 3x3 Matrix

To find the determinant of a 3x3 matrix, find the sum of the product of the elements of any of its row/column and their corresponding cofactors. Here is an example.

A = \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\). Let us use the first row to find the determinant.

det A = 1 (cofactor of 1) + 2 (cofactor of 2) + (-1) cofactor of (-1)
= 1(-3) + 2(-4) + (-1)5
= -3 - 8 - 5
= -16

But here is a trick to find the determinant of any 3x3 A = \(\left[\begin{array}{ccc}a & b & c \\ p & q & r \\ x & y & z\end{array}\right]\) matrix faster. Here, we just write the same matrix twice next to each other and then apply the trick.

The inverse of a 3x3 matrix formula uses the determinant of the matrix.

The inverse of a 3x3 matrix A is calculated using the formula A^-1 = (adj A)/(det A), where

• adj A = The adjoint matrix of A
• det A = determinant of A

det A is in the denominator in the formula of A^-1. Thus, for A^-1 to exist det A should not be 0. i.e.,

• A^-1 exists when det A ≠ 0 (i.e., when A is nonsingular)
• A^-1 does not exist when det A = 0 (i.e., when A is singular)

Thus, here are the steps to find the inverse of 3x3 matrix. The steps are explained through the same example A = \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\). Let us find A^-1.

• Step - 1: Find adj A.
  We have already seen that adj A = \(\left[\begin{array}{ccc}
  -3 & -4 & 5 \\
  -4 & 0 & -4 \\
  5 & -4 & -3
  \end{array}\right]\).
• Step - 2: Find det A.
  We have already seen that det A = -16
• Step - 3: Apply the inverse of 3x3 matrix formula A^-1 = (adj A)/(det A). i.e., divide every element of adj A by det A.
  Then A^-1 = \(\left[\begin{array}{ccc}
  -3/-16 & -4/-16 & 5/-16 \\
  -4/-16 & 0/-16 & -4/-16 \\
  5/-16 & -4/-16 & -3/-16
  \end{array}\right]\)
  = \(\left[\begin{array}{rr}3 / 16 & 1 / 4 & -5 / 16 \\
  1 / 4 & 0 & 1 / 4 \\
  -5 / 16 & 1 / 4 & 3 / 16 \\ \end{array}\right]\).

Like any other square matrix, we can use the elementary row operations to find the inverse of a 3x3 matrix as well. The process is explained below with an example.

• We first write the given 3x3 matrix A and the identity matrix I of order 3x3 as an augmented matrix separated by a line where A is on the left side and I is on the right side.
• Apply row operations so as to make the left side matrix to become the identity matrix I.
• Then the matrix on the right side is A^-1.

We can an see an example for this in the upcoming sections.

We can solve the system of 3x3 equations using the inverse of a matrix. The steps for this are explained here with an example where we are going to solve the system of 3x3 equations x + 2y - z = 10, 2x + y + 2z = 5, and -x + 2y + z = 6.

• Step - 1: Write the given system of equations as AX = B.
  \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\) \(\left[\begin{array}{rr}x \\y \\ z \end{array}\right]\) = \(\left[\begin{array}{rr}10 \\ 5 \\ 6 \end{array}\right]\)
  Here, A = \(\left[\begin{array}{rr}1 & 2 & -1 \\ 2&1&2 \\ -1 & 2&1 \end{array}\right]\), X = \(\left[\begin{array}{rr}x \\y \\ z\end{array}\right]\), and B = \(\left[\begin{array}{rr}10 \\ 5\\ 6 \end{array}\right]\).
• Step - 2: Find the inverse of the 3x3 matrix. i.e., find A^-1.
  In one of the previous sections, we found that A^-1 = \(\left[\begin{array}{rr}3 / 16 & 1 / 4 & -5 / 16 \\
  1 / 4 & 0 & 1 / 4 \\
  -5 / 16 & 1 / 4 & 3 / 16 \\ \end{array}\right]\).
• Step - 3: Find the solution matrix X using the formula X = A^-1B.
  X = \(\left[\begin{array}{rr}3 / 16 & 1 / 4 & -5 / 16 \\
  1 / 4 & 0 & 1 / 4 \\
  -5 / 16 & 1 / 4 & 3 / 16 \\ \end{array}\right]\) \(\left[\begin{array}{rr}10 \\ 5 \\ 6 \end{array}\right]\)
  = \(\left[\begin{array}{rr}5/4 \\4 \\ -3/4 \end{array}\right]\)

Therefore, x = 5/4, y = 4, and z = -3/4 is the solution of the given system of equations.

Important Notes on Inverse of 3x3 Matrix:

• A matrix A is invertible (inverse of A exists) only when det A ≠ 0.
• If A and A^-1 are the inverses of each other, then AA^-1 = A^-1A = I.
• The inverse of a 3x3 identity matrix is itself. i.e., I^-1 = I.
• The inverse of 3x3 matrix is used to solve a system of 3x3 equations in 3 variables.

☛ Related Topics:

• Inverse Matrix Calculator
• Matrix Multiplication
• Inverse of 2x2 Matrix
• Matrix Formula

What is Meant by Inverse of a 3x3 Matrix?

The inverse of a 3x3 matrix A is denoted by A^-1. Here, AA^-1 = A^-1A = I, where I is the identity matrix of order 3x3.

How to Find Inverse of a 3x3 Matrix?

Here are the steps to find the inverse of a 3x3 matrix A:

• Find det A.
• Find adj A.
• Apply the formula A^-1 = (adj A)/(det A).

What is an Example of a 3x3 Matrix with No Inverse?

A matrix cannot have inverse if its determinant is 0. A = \(\left[\begin{array}{rr}1 & 2 & 1 \\ 2&4&2 \\2 & 4 &5 \end{array}\right]\) has no inverse as det A = 0 in this case.

Are All 3x3 Matrices Invertible?

No, all 3x3 matrices are not invertible as a matrix cannot have its inverse when its determinant is 0. For example, A = \(\left[\begin{array}{rr}0 & 0 & 0 \\ -1&3&2 \\5 & 7 &5 \end{array}\right]\) is not invertible as det A = 0 in this case.

What is the Inverse of 3x3 Matrix Formula?

If A is a 3x3 matrix, its inverse formula is A^-1 = (adj A)/(det A). Here,

• det A = Determinant of the matrix A
• adj A = Adjoint of the matrix A

Does a 3x3 Matrix have an Inverse?

A 3x3 matrix has inverse only if its determinant is not zero. If the determinant is zero, then the matrix has is not invertible (does not have inverse) and in that case, it is called a singular matrix.

How to Find Inverse of a 3x3 Matrix Using Elementary Row Operations?

For finding the inverse of a 3x3 matrix (A ) by elementary row operations,

• Write A and I (identity matrix of same order) in a single matrix separating them by a vertical dotted line.
• Apply elementary row operations so that the left side matrix becomes I.
• The matrix that comes on the right side is A^-1.