Find the inverse, if it exists, for each matrix.

Question

[\begin{matrix} 2 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{matrix}]

Accepted Answer

Hello, everyone. We are asked to determine the inverse of the following matrix. If possible, we are given a three by three matrix. The first row reads 455, 2nd row 765, 3rd row 756. Recall that an inverse is found by doing one divided by the determinant of that matrix multiplied by the adjoint matrix. So we are gonna begin by finding the determinant of this three by three matrix. We do. So by multiplying each element of the first row by the minor matrix created when we exclude the rows and columns, that element is part of. So we have four multiplied by the minor matrix. Top row 65, 2nd row 56. We also alternate our positives and negatives four was positive. So next, we're going to subtract five and multiply it by the minor matrix that is formed when we exclude the rows and columns, that element is in. So that gives us a top row of 75, the second row of 76. And then now another positive. So plus five multiplied by the minor matrix top row 76, 2nd row 75, we are going to find the determinant of these two by two matrices. Recall we do that by diagonally multiplying left to right and then subtracting the product from right to left. So we get four multiplied by minus five, multiplied by seven plus five, multiplied by negative seven. So 44 minus 35 minus 35 our determinant here is negative 26. So we will need that again to finalize this inverse. But now we're gonna work on our adjoint matrix and to find our joint matrix, we first need to find a cofactor matrix. And so this will be formed by minor matrices that happen when we exclude the element um rows and columns. So what I mean the first element in our matrix is the number four. So we're gonna exclude the rows and columns that include that four. And we're left with the minor matrix 65 is the top row 56 is the second row. That's gonna be our first element. We're ultimately gonna find the determinant of all of these minor matrices. For our second element of the first row, we exclude the rows and columns including that number five. So we get a minor matrix with a first row of 75, second row of 76. And for our third element of that top row, we're going to exclude the five and the rows and columns it belongs to. And we get a minor matrix of 76 as the first row 75 is the second row into the second row of our Cofactor matrix. We're going to exclude the seven and the rows and columns it belongs to. So we get a minor matrix of 55 for the first row, 56 for the second row. Next element uh the number six in the second row of the original matrix. So we're excluding those rows and columns. So we get a minor matrix top row 45, 2nd row 76. And then third element here, we're gonna exclude the rows and columns including that five. The top row of the minor matrix is 45. 2nd row is 75, going to my third row, I exclude the element seven and it's rows and columns. So I get a minor matrix that the top row is 55. The second row is 56. Next, I'm excluding the five from the bottom row of the original matrix and the column it in rows it belongs to. So I get a minor matrix of 45 for the top row for the second row. And my last element, I'm excluding the six and its rows and columns. So I get a minor matrix. 45 is the top row 76 as the second row. Before we do anything else, I need to recall that I need to have alternating signs. So in my top row, we start positive, I need to put a negative in front of my minor matrix that's in the second spot, then positive. And then in my second row, that first element needs a negative sign or a minus sign. Second elements positive third elements negative in the third row, first elements positive, second elements negative third elements positive. So now we are gonna find the determinant of each of these minor matrices including the signs I just put in there. Again, we go left to right multiplying diagonally and subtract the product from right to left. And when we do so we get the top row is 11, negative seven, negative seven, second row, negative five, negative 11 15, third row, negative 5 15, negative 11. And we're almost there one more thing before we could say we have our adjoint matrix, we need to transpose the rows and columns from this cofactor matrix. And that'll make us our a joint matrix. So if it was a row before it's a column, so our first row which is 11 negative seven negative seven instead of writing it across it will now go down. So we have 11 and next element down is negative seven. Next element down after that is negative seven, our second row negative five under that negative 11 under that 15, our third row negative five 15 is under it and negative 11 is under that. So a joint matrix if we're reading the rows, the first row is 11 negative five negative five. The second row is negative seven, negative 11 15. The third row is negative 7 15, negative 11. Now we could do the part where we actually get to the inverse. So we're gonna do one divided by our determinant which we had negative 26. And we are multiplying that by our adjoint matrix we just wrote. So 11 negative five, negative five is my top row, negative seven, negative 11 15 is my second row, negative 7 15, negative 11 is my third row. Recall that when we multiply a scalar times a matrix, we multiply each element by the scalar. So by the number, so by negative 1 and when we do so we get negative 11 26 is our first term. Um 5 26 is our second term. 5 26 is again the third term in that first row. Second row we have 7 26th, 11 26th, negative 15, 26. Third row 7 26 negative 15, 26 and 11 26. If this could be reduced, we would reduce our fractions. However, none of these are reducible. So this is our inverse matrix have a nice day.

Find the inverse, if it exists, for each matrix. $[\begin{matrix} 2 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{matrix}]$

Key Concepts

Matrix Inverse

Determinant of a Matrix

Methods for Finding the Inverse

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