If A is any square matrix, then
Let's break it down: we verify A+At is symmetric by full calculation: use standard identity, substitute, simplify step by step to reach A+At is symmetric.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
#matrix theory
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If A and B are two matrices such that AB = A and BA = B, then which one of the following is correct?
Look at it this way: complete solution: write formula, plug data, do arithmetic. Final value matches (Aᵀ)² = Aᵀ. Simple and accurate.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Which of the following conditions holds true for a skew-symmetric matrix?
Understanding this is easy: using formula we get A = -A’. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get A = -A’.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Which of the following is the reversal law of transposes?
We can see that complete solution: write formula, plug data, do arithmetic. Final value matches (AB)’ = B’A’. Simple and accurate.
Ref: NCERT Mathematics Class XI and XII
For matrices of the same dimension M, N, and scalar c, which one of these properties DOES NOT ALWAYS hold?
Think of it like this: complete solution: write formula, plug data, do arithmetic. Final value matches MN = NM. Simple and accurate.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Consider two matrices M₁ and M₂ with M₁*M₂ = 0 and M₁ is non singular. Then which of the following is true?
Applying the basic concept, we verify M₂ is null matrix by full calculation: use standard identity, substitute, simplify step by step to reach M₂ is null matrix.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
If A and B are two n x n non singular matrix, then _____________
Recall the rule non-singular means |A|≠0,|B|≠0. Formula |AB|=|A||B|≠0. So AB also non-singular, inverse exists. Complete reasoning.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
