Practice Makes Perfect Algebra - Carolyn Wheater [37]
In order for two matrices to be called inverses, their product must be such an identity matrix. We know, too, that matrix multiplication is not generally commutative, so we add the condition that to be called inverses the two matrices must produce the same identity when multiplied in either order. That requires that the two matrices both be square. If they were not square, then the product A × B will be of a different size than the product B × A, even if both are identities.
If A and B are square matrices and I is an identity matrix of the same dimension and if A × B = B × A = I, then A and B are inverse matrices. We denote the inverse of matrix M as M−1.
When considering whether the inverse of a particular matrix exists, it is wise to first calculate the determinant of the matrix in question. Since only square matrices have determinants, this serves as a reminder that only square matrices can have inverses. Not all square matrices actually do have inverses, however, and for reasons we will see in a few moments, those matrices that have determinants of 0 have no inverse. We say that such a matrix is not invertible.
Verifying inverses
To determine whether two matrices are inverses, we must check both possible products. To determine if and are inverses, we check both the products A × B and B × A.
It is important to check both products. It is possible to find two matrices that produce an identity when multiplied in one order, but not in the other.
Matrices such as M and N in the example above are sometimes referred to as one-sided inverses, but they are not useful for any of the applications we will investigate.
One failure is enough to tell us that the two matrices are not inverses, however. If the first product we check does not yield an identity, we can stop and conclude that the matrices are not inverses without checking the other product. To determine if are inverses, we examine the product
Since this product is not an identity, we can conclude that the matrices are not inverses. It is not necessary to check the product .
Finding the inverse of a 2 × 2 matrix
Given two matrices, verifying whether they are inverses is a simple matter of multiplication. Most often, however, we have only one matrix and need to find its inverse if one exists. Finding the inverse of a 2 × 2 matrix is a relatively simple process, but for larger matrices, the process becomes more complex.
To find the inverse of a 2 × 2 matrix,
• Find the determinant of the matrix
• Exchange the elements on the major diagonal
• Change the signs of the elements on the minor diagonal
• Multiply by the reciprocal of the determinant
To find the inverse of the matrix , we first find the determinant of the matrix.
Remember that matrices with determinants of 0 have no inverse. Next the elements on the major diagonal, 1 and 2, exchange places, and the signs of the elements on the minor diagonal are changed. Of course, since 0 is neither positive nor negative, it remains 0.
Finally, we perform a scalar multiplication, multiplying by the reciprocal of the determinant, . In cases when the determinant is 0, it is impossible to find a reciprocal and the process is stopped.
The inverse of the matrix is the matrix .
Finding the inverse of a larger matrix
The steps outlined above are useful only for 2 × 2 matrices. For larger matrices, we will generally rely on the help of calculators, but it is worthwhile to explore the process a bit, at least for the 3 × 3 matrix. For such a matrix, the steps in finding the inverse, if it exists, are
• Find the determinant of the matrix
• Fill a matrix of the same size with position signs
• Determine the cofactor of each element and fill the matrix of cofactors
• Transpose the matrix of cofactors to form the adjoint
• Multiply the adjoint by the reciprocal of the determinant
For our exploration, we will