Inverting a Matrix Basic Math for Economics – Refresher 1 Introduction    With an understanding of basic matrix algebra and determinants, we are ready to discuss how to invert a matrix. Recall that matrix inversion is a useful technique that allows us to solve systems of equations directly.  This technique is commonly used when calculating regressions in econometrics. Inverting a matrix involves four steps: Calculate minors. Reflect across the trace. Adjust signs. Divide by the determinant. 2 Matrix Inversion Suppose we wanted to invert the following matrix, 𝑎 𝑏 𝐴= 𝑐 𝑑  Let’s follow our steps.   First, we need to calculate the minors. In words, we need to calculate the determinant for the submatrix that encompasses all elements that are not in the same row or column as our current element. For a 2x2 matrix, this is actually quite simple. 3 Matrix Inversion  Suppose we wanted to calculate the minor for our first element of the inverted matrix.  We need to look at the determinant for every element that is not in the first row or the first column, 𝑎 𝑏 𝐴= 𝑐 𝑑  This leaves us with just 𝑑, and the determinant of a single element in a matrix is just that element, so the element in the first row and first column of our inverted matrix is just 𝑑. 𝑑 ? ? ? 4 Matrix Inversion   We can continue this process for the other elements to obtain 𝑑 𝑐 𝑏 𝑎 Now we need to reflect across the trace.  (recall matrix) that the trace is just all of the diagonal elements of a  Effectively, we need to swap the element in row 1 and column 2 with the element in row 2 and column 1, etc. This updates our inverted matrix to: 𝑑 𝑏 𝑐 𝑎 5 Matrix Inversion  Adjust signs.  We need to multiply each term of the matrix by −1𝑟+𝑐 where 𝑟 is the row of the element and 𝑐 is the column of the element. Note that this coefficient can only take two values: 1 and − 1, depending on whether 𝑟 + 𝑐 is even or odd. The element in the first row and the first column of the matrix is multiplied by −11+1 = −12 = 1, and so on.  Updating our inverted matrix: 𝑑 −𝑏 −𝑐 𝑎 6 Matrix Inversion  Divide by the determinant.  Lastly, we divide by the determinant of the original matrix. Recall that the original matrix is, 𝑎 𝑏 𝐴= 𝑐 𝑑 and the formula for the determinant is, 𝐴 = 𝑎𝑑 − 𝑏𝑐  Since this is a scalar, we can just divide each individual term in the matrix by this number, yielding our final inverted matrix of 𝑑 −𝑏 − 𝑏𝑐 𝑎𝑑 − 𝑏𝑐 𝐴−1 = 𝑎𝑑−𝑐 𝑎 𝑎𝑑 − 𝑏𝑐 𝑎𝑑 − 𝑏𝑐 7 Matrix Inversion  Let’s do this again, but with a 3x3 matrix.  Consider the following matrix, 2 6 4 𝐵= 1 8 3 5 7 9  Again we just need to follow the four steps to calculate our inverse matrix: Calculate minors. Reflect across the trace. Adjust signs. Divide by the determinant. 8 Matrix Inversion  Calculate minors.  This is a bit more complicated for our 3x3 matrix. Let’s look at the minor for the element in the first row, second column (6).  Recall that we need to calculate the determinant of the submatrix that contains every element neither in this row nor column, i.e., 2 6 4 𝐵= 1 8 3 5 7 9  This leaves us with a 2x2 matrix, 1 3 5 9 whose determinant is 1 ∗ 9 − 3 ∗ 5 = −6 9 Matrix Inversion Thus, the minor for the first row and second column of our inverted matrix is -6, ? −6 ? ? ? ? ? ? ?  Calculating the rest of our minors, 51 −6 −33 26 −2 −16 −14 2 10 We are now ready to move on to our next step.  10 Matrix Inversion 51 26 −14  Reflect across the trace. −6 −33 −2 −16 2 10  This is the same as in our 2x2 example; we just mirror the elements of the matrix across its diagonal. Updating our inverse matrix, 51 26 −14 −6 −2 2 −33 −16 10 11 Matrix Inversion 51 26 −6 −2 −33 −16  −14 2 10 Adjust signs.  Once again, we multiply each term by −1𝑟+𝑐 to adjust our signs as need be. Updating our inverse matrix, 51 −26 −14 6 −2 −2 −33 16 10 12 Matrix Inversion  Divide by the determinant.  Lastly, we divide by the determinant, which we calculate from the original matrix, 2 6 4 𝐵= 1 8 3 5 7 9  Recall the formula for the determinant of a 3x3 matrix, 𝐵 =2 8 9 +6 3 5 +4 1 7 −2 3 7 −6 1 9 −4 8 5 =6  Thus, we divide each term in our matrix by 6 to obtain our inverted matrix. 13 Matrix Inversion 51 −26 −14 6 −2 −2 −33 16 10  Dividing each term by 6, we have our inverted matrix, 8.5 −4.33 −2.33 𝐵−1 = 1 −0.33 −0.33 −5.5 2.67 1.67  We can use this technique for a matrix of any size. Note that as matrices become larger, the calculations involved become more complicated. 14

Inverting a Matrix

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock