12.3 Inverse Matrices


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1 2.3 Inverse Matrices Two matrices A A are called inverses if AA I A A I where I denotes the identit matrix of the appropriate size. For example, the matrices A A If we think of the identit matrix I as being analogous to the number, then inverses are analogous to a reciprocal pair of numbers, such as 2/3 3/2. are inverses, since AA A A I I Here are some of the most important properties of inverse matrices:. The inverse of a matrix is unique. If A is a matrix, then there can be onl one possible inverse A for A. To underst wh, suppose that A is an inverse for A, suppose that B is also an inverse for A. Then You should be able to underst each step of this long equation. B BI BAA IA A so B must be the same matrix as A. 2. The inverse of the inverse is the original matrix. The relationship between A A is smmetric, so the inverse of A is the original matrix A. ( A ) A Nonsquare matrices whose product is the identit are sometimes called onesided inverses. For square matrices, if the product AB is the identit matrix, then BA is alwas the identit as well, so A B are inverses of one another. 3. Onl square matrices can have inverses. Though it is possible for the product of two nonsquare matrices to be the identit matrix if we reverse the order of multiplication we no longer get the identit Onl for square matrices can both products ields the identit matrix.
2 INVERSE MATRICES 2 4. Not ever square matrix is invertible. Some square matrices do not have an inverse. For example, the zero matrix Sometimes invertible matrices are referred to as nonsingular. certainl has no inverse, since the product of the zero matrix with an other matrix is alwas the zero matrix. there are also less obvious examples, such as the matrix which has no inverse. In general, a square matrix is called invertible if it has an inverse, singular if it does not. 5. The product of invertible matrices is invertible. If A B are invertible matrices, then the product AB is invertible as well, with (AB) B A Note that the order of the two matrices is reversed when we take the inverse. This order reversal is necessar to make the inverses cancel with AB when multiplied. In particular, ABB A AIA AA I B A AB B IB B B I Finding the Inverse There is a simple formula for the inverse of a 2 2 matrix: a b c d ad bc d c b a It is eas to check that this formula works. In particular, ad bc d c b a b a c d ad bc 0 ad bc 0 ad bc 0 0 EAMPLE Find the inverse of the matrix According to the formula above, the inverse is (3)(6) (2)(8) /2 Incidentall, the denominator ad bc in this formula is called the determinant of the 2 2 matrix. A 2 2 matrix whose determinant is zero is not invertible.
3 INVERSE MATRICES 3 Finding the inverse of a 3 3 or larger matrix is much more difficult. For example, the formula for the inverse of a 3 3 matrix is a b c d e f g h i aei a f h + b f g bdi + cdh ce g ei f h ch bi b f ce f g di ai c g cd a f dh e g b g ah ae bd This formula is so complicated that it is rarel worthwhile to compute the inverse of a 3 3 matrix b h, let alone larger sizes such as 4 4 or 5 5. Fortunatel, most modern calculators computer algebra sstems can compute the inverse of a matrix, several websites offer tools that let ou perform matrix computations online. Algebra of Inverse Matrices Inverse matrices let ou divide both sides of an equation b a matrix. For example, if we have a matrix equation of the form A B where is an unknown matrix, we can solve for b mutlipling both sides of the equation b A : A A A B Since A A I, the left side simplifies to just, so A B EAMPLE 2 Find a 2 2 matrix so that Here is the identit matrix, so is the same as I, which is just. which simplifies to so Multipling through b the inverse of gives (4)(2) (6)() 4 3 / /2 2 B the wa, since the order of matrix multiplication matters, one must be careful to multipl b A in the same wa on both sides of an equation. Given an equation Y
4 INVERSE MATRICES 4 we can either leftmultipl b A on both sides A A Y or we can rightmultipl b A on both sides A YA However, we cannot leftmultipl b A on one side of the equation rightmultipl b A on the other side. EAMPLE 3 Make sure to underst the difference between this example the previous one. Find a 2 2 matrix so that This time we must rightmultipl both sides b the inverse of 4 6. This gives which simplifies to / , so / /2 2 Solving Linear Sstems with Inverses Recall that an n n linear sstem can be written in the form Ax b where A is the (square) matrix of coefficients, b is the vector of constant terms. If we know the inverse of A, we can solve this linear sstem b leftmultipling both sides of the equation b A. This gives x A b Thus ou can solve a linear sstem b finding the inverse of the matrix of coefficients. This can be quite useful if ou have several linear sstems with the same coefficients but different constant terms. EAMPLE 4 Solve each of the following linear sstems: 3x x x + 2 5x x x + 3 5
5 INVERSE MATRICES 5 Each of these sstems is reall a vector equation: x x Thus the solutions are respectivel x x so the solutions are x (3)(3) (2)(5) x 3 2 x x x EERCISES 4 Find the inverse of the given matrix, or state that the matrix is not invertible Find scalars a, b, c so that the given pair of matrices are inverses , a b 2 3 c , 3 8 a 7 b 6 c 2 7. Suppose that A, B, C are invertible n n matrices. (a) Is the product ABC invertible? If so, what is its inverse? (b) Is AB C invertible? If so, what is its inverse? 8 Find a 2 2 matrix that satisfies the given equation Use inverse matrices to find the solution to the each of the following linear sstems: 4x x x x x x + 3
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