16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-155 Finding the inverse of a matrixThe 2 × 2 matrix A is invertible if and only if det(A) ≠ 0.LettingA =[ ]a11 a 12,a 21 a 22we can compute A −1 from the formulaA −1 =Example 5.Find the matrix X such that[2 1−1 2[1det(A)]X =]a 22 −a 12.−a 21 a 11[3 0−2 1Here the matrix X is itself 2 × 2. This has the form A · X = B for 2 × 2matrices.If A is non-singular, then we get the answer fromX = A −1 B.We check non-singularity by computing det(A). We get det(A) = 5, sothe matrix is non-singular and its inverse is(1/5)[2 −11 2]]so,X = A −1 B = (1/5)[2 −11 2] [3 0−2 1]=[8/5 −1/5−1/5 2/5]For higher dimensional matrices, the formula for the inverse is harder, soit is convenient to find another method.Suppose that the dimension of A is n.Then, we are looking for an n × n matrix B such that A · B = I where Iis the n × n identity matrix.