Which of the following statement about the matrix is incorrect?

Question

Options

• AThe arbitrary matrix is multiplied by the unit matrix and will not change.
• BThe transpose of the symmetric matrix A 1s also A itself.
• CThe transpose of the orthogonal matrix A is equal to the inverse of A .
• DThere is no inverse matrix for the diagonal matrix.

• A
  4% (2)
• B
  15% (7)
• C
  7% (3)
• D
  74% (34)

Explanation

Option D is incorrect because diagonal matrices can have inverses - specifically, a diagonal matrix is invertible whenever all of its diagonal entries are non-zero. The inverse of such a matrix is simply the diagonal matrix formed by taking the reciprocal of each diagonal entry (e.g., a diagonal entry of 3 becomes 1/3). Only a diagonal matrix with at least one zero on the diagonal is non-invertible (singular).

The other three statements are all true and serve as distractors: A correctly describes the identity (unit) matrix property - multiplying any matrix by it leaves the matrix unchanged; B correctly states the definition of a symmetric matrix, where A^T = A by definition; C correctly defines an orthogonal matrix, where A^T = A⁻¹, which is equivalent to A^T A = I.

Memory tip: Use the identity matrix as your anchor - it is a diagonal matrix (all 1s on the diagonal), and it's clearly invertible (it's its own inverse). If even one diagonal matrix has an inverse, the blanket claim in D must be false.

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