“What is the determinant of the identity matrix?” Here’s a breakdown:
* Identity Matrix: The identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. For example, a 2×2 identity matrix looks like this:
“`
[1 0]
[0 1]
“`
And a 3×3 identity matrix looks like this:
“`
[1 0 0]
[0 1 0]
[0 0 1]
“`
* Determinant: The determinant is a scalar value that can be computed from the elements of a square matrix. It provides information about the matrix, such as whether the matrix is invertible (has an inverse).
* Why the determinant is 1:
* For a 2×2 matrix [a b; c d], the determinant is calculated as (a*d) – (b*c). For the 2×2 identity matrix, this is (1*1) – (0*0) = 1.
* For larger identity matrices, the determinant can be found using various methods (e.g., cofactor expansion), but the result will always be 1. This is because the identity matrix represents a transformation that doesn’t change the space, so its determinant (which reflects the scaling factor of the transformation) is 1.
So, the key detail is understanding what an identity matrix is and knowing that its determinant is a fundamental property.
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