lec2_3

lec2_3 - 0.00000 0.00000 1.00000 -2.00000 -3.00000 1.00000...

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NOTES ON SECTION 2 . 3 MA265, SECTIONS 41, 52 Key words 1. Elementary matrices It appears that we will not discuss elementary matrices directly, but you may as well have a look over this section in the textbook yourself. 2. Finding A - 1 We present an algorithm for computing A - 1 , which is easily described. Theorem 1. The matrix A is invertible if and only if the augmented matrix [ A | I ] (formed by augmenting the matrix A with an identity matrix of the same size) has RREF of the form [ I | B ] (where I is again the identity matrix). If so, the matrix B is in fact the inverse B = A - 1 . Corollary 2. A matrix A is singular if and only if its RREF has a row of zeros. Example 3. Find 1 2 - 3 1 - 2 1 5 - 2 - 2 - 1 if it exists, and if it doesn’t explain why. Find the RREF of [ A | I ] – we will do this using MATLAB with the commands >>> A = [1,2,-3;1,-2,1;5,-2,-2] A = 1 2 -3 1 -2 1 5 -2 -2 >>> rref([A,eye(3)]) The output is ans = 1.00000 0.00000 0.00000 -1.50000 -2.50000 1.00000 0.00000 1.00000 0.00000 -1.75000 -3.25000 1.00000
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Unformatted text preview: 0.00000 0.00000 1.00000 -2.00000 -3.00000 1.00000 which tells us that the inverse is A-1 = -1 . 5-2 . 5 1-1 . 75-3 . 25 1-2-3 1 Example 4. Find 1 2-3 1-2 1 5-2-3 -1 if it exists, and if it doesnt explain why. 1 Do the same thing, but this time the RREF is ans = 1.00000 0.00000 -1.00000 0.00000 -0.25000 0.25000 0.00000 1.00000 -1.00000 0.00000 -0.62500 0.12500 0.00000 0.00000 0.00000 1.00000 1.50000 -0.50000 Since the left side of the partition is NOT an identity matrix (note the row of zeros!) we conclude that A is not invertible. 3. A fact about singular matrices Theorem 5. A matrix is singular if and only if the homogeneous system Ax = 0 has a non-trivial solution. This observation is important later (e.g. when were trying to gure out how to nd eigenvalues and eigenvectors). 2...
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lec2_3 - 0.00000 0.00000 1.00000 -2.00000 -3.00000 1.00000...

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