Consider the matrices 𝐴 and 𝐵. Determine the inverse of 𝐴 plus 𝐵.
Much like the order of operations, sometimes called PEMDAS or BIDMAS, we should begin here by completing the part inside the parentheses. That’s the sum of 𝐴 and 𝐵. We can add matrices of the same size — that’s here two-by-two matrices — by simply adding the individual elements.
To get the element in the first row and first column, we’ll add negative three and negative one. That’s negative four. To get the element in the first row and second column, we’ll add negative two and two, which is zero. We can repeat this process for the remaining two elements. Negative five plus eight is three, and negative seven plus nine is two. The sum of 𝐴 and 𝐵 is therefore negative four, zero, three, two.
The next step is to find the inverse of this matrix. Remember, for a two-by-two matrix 𝐴 with elements 𝑎, 𝑏, 𝑐, 𝑑, the inverse of 𝐴 is one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant of 𝐴 is found by multiplying the elements 𝑎 and 𝑑 and subtracting the product of the elements 𝑏 and 𝑐. Notice that this means if the determinant of the matrix is zero, then there can be no multiplicative inverse, since one over the determinant of 𝐴 is one over zero, which we know to be undefined.
Let’s begin by calculating the determinant of 𝐴 plus 𝐵. We multiply the top left and bottom right elements. And we subtract the product of the top right and bottom left. That gives us the determinant of negative eight. Since this determinant is not equal to zero, we can indeed find the inverse.
Let’s substitute what we know about the matrix 𝐴 plus 𝐵 into the formula for the inverse. One over the determinant is one over negative eight or negative one-eighth. And if we label 𝑎 to be negative four, 𝑏 to be zero, 𝑐 to be three, and 𝑑 to be two, we get two, zero, negative three, negative four.
Finally, we need to multiply each element by negative one-eighth. Negative one-eighth multiplied by two is negative a quarter or negative 0.25. Negative eighth multiplied by zero is zero. A negative multiplied by a negative is a positive. So negative eighth multiplied by negative three is 0.375. And negative one-eighth multiplied by negative four is 0.5 or a half.
The inverse of 𝐴 plus 𝐵 is negative 0.25, zero, 0.375, and 0.5.