Video: Finding the Inverse of a matrix Using the Properties of Determinants

Consider the matrix (1, 0, 3 and 1, 0, 1 and 3, 1, 0). Determine whether the matrix has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the formula for the inverse which involves the cofactor matrix.

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Video Transcript

Consider the matrix. Determine whether the matrix has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse by using the formula for the inverse which involves the cofactor matrix.

In order to find the determinant of any matrix, we can use the following formula. We have that the determinant of this matrix here is equal to π‘Ž times by 𝑒𝑖 minus π‘“β„Ž minus 𝑏 times by 𝑑𝑖 minus 𝑓𝑔 plus 𝑐 lots of π‘‘β„Ž minus 𝑔𝑒. Now, let’s call the matrix given in the question 𝐴. And we need to find the determinant of 𝐴 or det 𝐴. Using the formula for the determinant, we get that det 𝐴 is equal to one multiplied by zero times zero minus one times one minus zero multiplied by one times zero minus one times three plus three multiplied by one times one minus three times zero.

Here, we have a lot of things which are multiplied by zero. And anything which is multiplied by zero will be zero. So when we add or subtract it, it has no effect. Therefore, anything which is multiplied by zero here we can cross out. What this leaves us with is one times by minus one plus three times by one. And so this is equal to negative one plus three, which gives us two. Therefore, our determinant is nonzero. And so our matrix has an inverse.

Let’s first recall that the formula for the inverse of a matrix including the cofactor matrix is that the inverse matrix is equal to one over the determinant of the matrix multiplied by the matrix of cofactors. Since we’ve already found the determinant of 𝐴, we just need to find the matrix of cofactors of 𝐴. In order to do this, we first need to find the matrix of minors of 𝐴, which we will call 𝐴 𝑀. To find each entry in the matrix of minors, we simply cross out the row and column of the corresponding entry on the original matrix and find the determinant of the remaining two-by-two matrix.

Starting with the top left entry in 𝐴𝑀, we simply cross out the first row and the first column in 𝐴. And the entry in the matrix of minors is simply the determinant of the remaining two-by-two matrix, which looks like this. For the second entry, we simply cross out the first row and the second column. And the entry is the determinant of the remaining two-by-two matrix, which is this. For the third entry, we cross out the first row and the third column, leaving us with a two-by-two matrix. And the third entry in the matrix of minors is simply the determinant of this. We continue doing this for each entry in the matrix of minors. This is what we are left with.

And we can use the following formula in order to find these determinants. We have that the determinant of π‘Ž, 𝑏, 𝑐, 𝑑 is equal to π‘Žπ‘‘ minus 𝑏𝑐. So the first entry in that matrix of minors is equal to zero times zero minus one times one, which is simply negative one. The second entry is equal to one times zero minus one times three, which gives us negative three. The third entry is equal to one times one minus zero times three, which is just one. And we can carry on like this for the rest of the matrix. So now, we have found our matrix of minors.

In order to get the matrix of cofactors from this matrix of minors, we simply need to change some of the signs in the matrix of minors. This matrix here represents which signs need to change in the matrix of minors in order for it to become the matrix of cofactors. In our case, we need to change the signs of negative three, negative three, one, and negative two. This will give us that our matrix of cofactors of 𝐴 or 𝐴 𝑐 is equal to negative one, three, one, three, negative nine, negative one, zero, two, zero.

Now that we have our matrix of cofactors of 𝐴, we’re able to find the inverse of 𝐴. Using the formula for the inverse of a matrix, we have that the 𝐴 inverse is equal to one over two multiplied by the matrix of cofactors, which we just found. And so this is the solution to the problem.

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