Consider the matrices 𝐴 equals 10, negative six, negative four, negative seven, 𝐴𝐵 equals 24, 72, 28, and negative 57. Find the matrix 𝐵.
We know that matrix 𝐴 times matrix 𝐵 equals the product matrix 𝐴𝐵. And we’re trying to find 𝐵. To find out what matrix 𝐵 is, we need to isolate it. We need to get it by itself. We do that by multiplying by the inverse matrix 𝐴 on both sides of the equation. The inverse matrix 𝐴 times matrix 𝐴 will give us an identity matrix that leaves only 𝐵 on the left side. And that means the matrix 𝐵 is equal to the inverse matrix 𝐴 multiplied by the product matrix we already have, 𝐴𝐵.
The inverse of a two-by-two matrix: if we start with the matrix 𝑎, 𝑏, 𝑐, 𝑑, its inverse is the determinate one over 𝑎𝑑 minus 𝑏𝑐. Then we flip the positions of 𝑎 and 𝑑 and then take negative 𝑏 and negative 𝑐. The inverse of 𝑎 will be the inverse of 10, negative six, negative four, negative seven. The determinate will be one over 10 times negative seven, negative 70, minus negative six times negative four, which is positive 24. Negative 70 minus 24 equals negative 94.
Our determinate is negative one ninety fourths, negative one over 94. We change the positions of negative seven and 10. We need to take the negative value of negative six, positive six. And the negative value of negative four is positive four. Our 𝐴 inverse matrix is negative one ninety fourths, negative seven, six, four, 10. Remember that, to find matrix 𝐵, we’re taking the matrix of 𝐴 inverse and multiplying it by 𝐴𝐵. To multiply these two two-by-two matrices together, we’ll find the dot product of the first row and the first column. That means we’ll multiply negative seven times 24. And then we’ll add six times 28. Negative seven times 124, which equals negative 168. Six times 28 equals positive 168. When we add those together, we get zero.
Moving on to the next position, we’ll take the dot product of the first row and the second column: negative seven times 72 plus six times negative 57. Negative seven times 72 equals negative 504. Six times negative 57 equals negative 342. The sum of these two values is negative 846.
And we’re ready to move to the next position, the second row, the first column. The dot product will be four times 24 plus 10 times 28. Four times 24 equals 96. 10 times 28 equals 280. And together they equal 376. Now the last position, second row, second column, four times 72 plus 10 times negative 57. Four times 72 equals 288. 10 times negative 57 equals negative 570. Together these two values equal negative 282. This finishes our matrix multiplication.
But we still have this scalar that we need to multiply through by. We must multiply every value in the matrix by negative one over 94. Zero times negative one over 94 equals zero. Negative one over 94 times negative 846 equals 846 over 94. The fraction becomes positive. 376 times negative one over 94 is negative 376 over 94. Negative one over 94 times negative 282 becomes positive 282 over 94. 846 over 94 reduces to nine. Negative 376 over 94 reduces to negative four. 282 over 94 can be reduced to three. The value of matrix 𝐵 is zero, nine, negative four, three.