It didn't look as neat as the previous solution, but it does show us that there is more than one way to set up and solve matrix equations. Example 2A: Using Cramer’s Rule for Two Equations Use Cramer’s rule to solve each system of equations. In fact it is just like the Inverse we got before, but Transposed (rows and columns swapped over). Then (also shown on the Inverse of a Matrix page) the solution is this: The rows and columns have to be switched over ("transposed"): I want to show you this way, because many people think the solution above is so neat it must be the only way.Īnd because of the way that matrices are multiplied we need to set up the matrices differently now. Quite neat and elegant, and the human does the thinking while the computer does the calculating.įor fun (and to help you learn), let us do this all again, but put matrix "X" first. Just like on the Systems of Linear Equations page. Then multiply A -1 by B (we can use the Matrix Calculator again): (I left the 1/determinant outside the matrix to make the numbers simpler) It means that we can find the values of x, y and z (the X matrix) by multiplying the inverse of the A matrix by the B matrix.įirst, we need to find the inverse of the A matrix (assuming it exists!) 4x4-16x20 Three solutions were found : x 2 x -2 x 0 Step by step solution : Step 1 :Equation at the end of step 1 : (4 (x4)) - 24x2 0 Step 2 :Equation at the end of step 2. Then (as shown on the Inverse of a Matrix page) the solution is this: A is the 3x3 matrix of x, y and z coefficients.Which is the original left side of our equations above (you might like to check that).
Why does go there? Because when we Multiply Matrices the left side becomes:
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