Question Video: Multiplication of Two Matrices Involving the Identity Matrix Mathematics

Given that 𝐴 = [βˆ’1, 4, and βˆ’1, 11] and 𝐼 is the identity matrix of the same order as 𝐴, find 𝐴 Γ— 𝐼 and 𝐼².

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Video Transcript

Given that matrix 𝐴 is equal to negative one, four, negative one, 11 and 𝐼 is the identity matrix of the same order as 𝐴, find 𝐴 multiplied by 𝐼 and 𝐼 squared.

We can begin by recalling that the identity matrix is a square matrix with ones along the leading diagonal and zeros everywhere else. In this case, the two-by-two identity matrix has elements one, zero, zero, one. In the first part of this question, we need to multiply the identity matrix by matrix 𝐴. When multiplying two matrices, we multiply the elements in each row of the first matrix by each column in the second matrix. We begin by multiplying the elements in the first row of the first matrix by the first column in the second matrix. This gives us negative one multiplied by one plus four multiplied by zero.

Next, we have negative one multiplied by zero plus four multiplied by one. We then repeat this process using the elements in the second row of the first matrix. We have negative one multiplied by one plus 11 multiplied by zero, and the element in the bottom right is negative one multiplied by zero plus 11 multiplied by one. This leaves us with the matrix negative one, four, negative one, 11. Matrix 𝐴𝐼 is the same as matrix 𝐴. This leads us to the general rule that matrix 𝐴 multiplied by the identity matrix is equal to matrix 𝐴. When dealing with the identity matrix, this relationship is commutative. 𝐴 multiplied by 𝐼 is equal to 𝐼 multiplied by 𝐴, which is equal to matrix 𝐴.

The second part of our question asked us to calculate the matrix 𝐼 squared. We need to multiply the matrix one, zero, zero, one by the matrix one, zero, zero, one. The top-left element will be equal to one as we have one multiplied by one plus zero multiplied by zero. In the top right, we have zero. We’re multiplying the elements in the first row of the first matrix by the second column in the second matrix.

Repeating this process gives us the element zero and one in the second row. 𝐼 squared is equal to one, zero, zero, one, which is the identity matrix. We can therefore conclude that 𝐼 squared is equal to 𝐼. This leads us to a second general rule involving matrix multiplication. Raising the identity matrix to any power gives us the identity matrix. If matrix 𝐴 equals negative one, four, negative one, 11, then 𝐴𝐼 is equal to 𝐴 and 𝐼 squared is equal to 𝐼.

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