Given that matrix 𝐴 is equal to negative one, five, zero, five, matrix 𝐵 is equal to five, negative five, zero, negative one, and 𝐼 is the unit matrix of the same order, find 𝑥 for which 𝐴𝐵 is equal to 𝑥 multiplied by 𝐼.
In this question, we need to calculate the scalar or constant 𝑥 such that matrix 𝐴 multiplied by matrix 𝐵 is equal to 𝑥 multiplied by matrix 𝐼. We recall that for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this question, both 𝐴 and 𝐵 are two-by-two square matrices. We can therefore calculate the matrix 𝐴𝐵 by multiplying negative one, five, zero, five by five, negative five, zero, negative one.
When multiplying two matrices, we multiply the elements in the rows of the first matrix by the columns of the second matrix. Multiplying the first row of matrix 𝐴 by the first column of matrix 𝐵 gives us negative one multiplied by five plus five multiplied by zero, which is equal to negative five. We then multiply the first row of matrix 𝐴 by the second column of matrix 𝐵. This gives us zero. We can then repeat this process with the second row of matrix 𝐴, giving us elements zero and negative five. The matrix 𝐴𝐵 is equal to negative five, zero, zero, negative five.
We are also told in the question that 𝐼 is the unit matrix of the same order. This means that 𝐼 is equal to one, zero, zero, one. The unit or identity matrix has ones on its main or leading diagonal and zeros elsewhere. Our final step is to calculate the constant 𝑥 such that negative five, zero, zero, negative five is equal to 𝑥 multiplied by one, zero, zero, one. When multiplying any matrix by a scalar or constant, we simply multiply each of the elements or components in the matrix by that constant. The matrix negative five, zero, zero, negative five is equal to negative five multiplied by the matrix one, zero, zero, one. This means that the value of 𝑥 is negative five.