Since the product of A A A and B B B is an identity matrix, then that means A A A and B B B are two matrices inverses of each other. Just as done in problem example 1, use the two matrices ( A A A and B B B) defined below and show if they are inverses of each other. Therefore, let us use that expression and multiply matrices X X X and Y Y Y in order to see what they produce:Īs you can see, the product of matrices X X X and Y Y Y happens to be the identity matrix of second order, therefore, these two matrices are inverses of each other. We attack this problem by remembering that two matrices are inverses of each other they will produce an identity matrix of their same dimensions when being multiplied, just as described in equation 2 for the case of 2x2 matrices. Using the matrices X X X and Y Y Y provided below: So, we will be looking into a method using row operations because we believe is a much more practical approach.įor now, we will continue to focus on the inverse of 2x2 matrices only, so let us continue.Įxercise problems for finding the inverse of a 2x2 matrix The reason is that we could use a process to find the inverse of a larger matrix based on the same principle as the one we are using today for a 2x2 matrix (which would also contain a factor of one over the determinant of the matrix), BUT, as the original matrix gets bigger, such process becomes too large and time consuming to be practical. Notice we have not used row operations during the calculation of a 2x2 inverse, so, why trying it with a different method when the matrix gets a little bigger?
INVERSE OF A MATRIX HOW TO
In later lessons we will look at how to compute the inverse of 3x3 matrices with matrix row operations. Remember the expression found in equation 1 provides de invertibility condition in general, meaning, it applies to square matrices of any order (dimensions).
Therefore we have proved that the expressions providing the conditions for invertibility of a matrix (shown in both equation 1 and equation 2), hold true. Hence, now we can finally prove the expression found in equation 2 by multiplying matrix A A A with is inverse.Įquation 10: Proving the condition of invertibility of AĪnd so, as you can see, the matrix multiplication between matrix A A A and its inverse produces the identity matrix with the same dimensions as them. Since our matrix A A A has a determinant which is not equal to zero, we can determine A A A is an invertible matrix and so, we can finally calculate its inverse. The first part of our proof is to verify this matrix is in fact an invertible matrix, for that, we obtain its determinant: For that we define matrix A A A as shown below: On this section we will prove how a 2x2 matrix and its inverse meet the condition defined in equation 2.
INVERSE OF A MATRIX SERIES
Later, in our last section, we will work through a series of exercises in order for you to practice. Having learnt the usage and how to get the inverse of a 2x2 matrix, let us go next into a section dedicated to prove that equations 2 and 5 are correct, if other words, let us calculate the inverse of 2x2 matrix proof with an example given matrix so you can observe the formula for inverse of 2x2 matrix in action. And so, we can conclude that B B B is equal to the inverse of A A A time C C C. Then, applying what we learnt in our lesson about the identity matrix, we know that any matrix multiplied by an identity matrix gives a result the non-identity matrix itself. In general, the condition of invertibility for a nxn matrix A A A is:Ī ⋅ A − 1 = A − 1 A ⋅ A = I n A \cdot A^ I n of the same dimensions as the original matrices. In other words, an invertible matrix is that which has an inverse matrix related to it, and if both of them (the matrix and its inverse) are multiplied together (no matter in which order), the result will be an identity matrix of the same order. In our past lesson we learnt that for an invertible matrix there is always another matrix which multiplied to the first, will produce the identity matrix of the same dimensions as them.