Symmetric and skew symmetric, and splitting any square matrix into one of each

What to assume they know

• The transpose, and what it does to an entry and to the order
• The four listed properties of the transpose, especially the sum law and the scaling law
• Matrix addition, subtraction and scaling
• Square matrices, the diagonal, and the zero matrix
• Matrix multiplication and the fact that two factors need not commute
• Equality of matrices, position by position
• Reading a statement of the form "this holds exactly when that holds"

What they should be able to do

• State the symmetric condition and the skew symmetric condition, each as one equation and as a condition on entries
• Derive, rather than assert, that a skew symmetric matrix has zeros on its diagonal
• Explain why only a square matrix can satisfy either condition
• Recognise both kinds from a printed matrix, including the letters-only instances the chapter uses
• Prove that a square matrix plus its transpose is symmetric, naming the rule used at each step
• Prove that a square matrix minus its transpose is skew symmetric
• Split a stated square matrix into a symmetric part and a skew symmetric part, and check that the two add back
• Say why the halving is necessary and what goes wrong without it
• Decide when a product of two symmetric matrices is itself symmetric
• Show that the only matrix satisfying both conditions is the zero matrix

Where it usually goes wrong

• "Symmetric means the matrix looks tidy." It means the entry at row i column j equals the entry at row j column i, for every pair. The chapter's own instance has a surd, a decimal and negative entries, and is symmetric.
• "Skew symmetric is just symmetric with minus signs everywhere." Only the off-diagonal pairs pick up the sign. The diagonal is forced to zero, and the chapter derives that rather than declaring it.
• "A skew symmetric matrix could have a non-zero diagonal if the entries were chosen cleverly." It could not, and the argument is three lines. Run it.
• "A rectangular matrix can be symmetric if it happens to look balanced." It cannot, because its transpose has a different order and the two cannot be compared at all. Section 5 should settle this from the order alone, before any entry is examined.
• "Theorem 2's halves are a convenient choice." They are the only choice. Without them the two pieces add to twice the original.
• "Splitting is only worth doing when the matrix is neither kind." Exercise 3.3 Q9 and Q10(ii) are the two boundary cases — one already skew, one already symmetric — and in both the recipe returns the right answer with a zero matrix for the other half.
• "The product of two symmetric matrices is symmetric." Only when they commute. Miscellaneous Example 24 is exactly this, and Exercise 3.3 Q11 is its companion.
• "The chapter proves these results because they are hard." It proves them because they follow from the transpose laws in three or four moves each. Naming the rule used at each step is the transferable skill here; the results themselves are short.

Questions to check understanding

• Classify a printed square matrix as symmetric, skew symmetric or neither, with a reason
• Show that the diagonal of a skew symmetric matrix must vanish
• Explain why neither condition applies to a matrix that is not square
• Prove that a square matrix plus its transpose is symmetric, naming the rule at each step
• Split a stated square matrix into its two parts and verify that they add back — the form of Example 22 and Exercise 3.3 Q10
• Split a matrix that is already one of the two kinds, and report the zero matrix for the other part — the form of Exercise 3.3 Q9 and Q10(ii)
• Decide the kind of the difference of two products of symmetric matrices — the form of Exercise 3.3 Q11 and Miscellaneous Exercise Q1
• Find the angle making a stated matrix satisfy a stated sum condition — the form of Exercise 3.3 Q12
• Show that a matrix of both kinds at once must vanish — the form of Miscellaneous Exercise Q10

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• Definition 4 and its instance (§3.6, Part I p. 63). The condition is stated both as one equation between the matrix and its transpose and as an equation between entries with the indices swapped. The instance is a three by three holding root three, two and three; two, minus one point five and minus one; three, minus one and one. Read the top-left entry off the page image — the text layer shows the radical as a bare three. Verified: the three off-diagonal pairs match across the diagonal, so the instance does satisfy the condition, and the diagonal is unconstrained, which is worth pointing at because the skew case is about to constrain it.
• Definition 5, and the derivation that follows it (§3.6, Part I p. 63). The condition is stated, then the chapter puts the two indices equal, obtains that a diagonal entry equals its own negative, doubles and concludes that every diagonal entry is zero. This is the first place in the chapter where a constraint on the entries is squeezed out of a definition rather than checked against an instance — §3.4.3 and §3.4.4 derive laws, but they derive them by descending to a position, which is a different move. Section 4 should slow right down here: three lines of algebra, one conclusion, no instances needed.
• The skew instance (Part I p. 64). A three by three built from three letters: zeros down the diagonal, three letters above it, and their negatives below. Verified: transposing turns each letter into its negative's position, so the condition holds identically in the letters, whatever values they take. This instance is more useful than a numerical one because it displays the general shape rather than one case of it.
• Theorem 1 and its proof (Part I p. 64). The claim: a square matrix plus its transpose is symmetric, and the same matrix minus its transpose is skew symmetric. The proof of the first half runs four steps, each annotated with the rule it uses — the sum law for the transpose, the double-transpose law, and the commutativity of matrix addition. The proof of the second half runs the same way and the chapter marks two of its steps with a parenthetical question instead of an annotation. Answer both aloud in the explanation: the first is the sum law applied to a difference, the second is the double-transpose law.
• Theorem 2 and its proof (Part I pp. 64–65). The claim is the splitting. The proof is two lines: write the matrix as half of the sum with its transpose plus half of the difference, then observe that the first bracket is symmetric and the second skew by Theorem 1, and that halving preserves both because scaling passes through a transpose. The scaling law is doing load-bearing work here and the chapter names it in the proof.
• Example 22 (Part I pp. 65–66). A three by three holding two, minus two and minus four; minus one, three and four; one, minus two and minus three, to be split. Verified: its transpose holds two, minus one and one; minus two, three and minus two; minus four, four and minus three. The symmetric part is two, minus three halves and minus three halves; minus three halves, three and one; minus three halves, one and minus three. The skew part is zero, minus one half and minus five halves; one half, zero and three; five halves, minus three and zero. The chapter then transposes each part to confirm its kind, and finally adds the two back to recover the original. Keep all four stages — build, check, build, check, add back. The add-back is the step students skip and it is the only one that shows the halves were necessary.
• Exercise 3.3 Q7 (Part I p. 67). Two matrices to be classified. Verified: the first, holding one, minus one and five; minus one, two and one; five, one and three, is symmetric. The second, holding zero, one and minus one; minus one, zero and one; one, minus one and zero, is skew symmetric — and its diagonal is zero, as the derivation of section 4 requires.
• Exercise 3.3 Q8 (Part I p. 67). A two by two holding one and five above six and seven. Verified: the sum with its transpose is two and eleven above eleven and fourteen, which is symmetric; the difference is zero and minus one above one and zero, which is skew. This is Theorem 1 on the smallest interesting case and it belongs in the explanation immediately after the proof.
• Exercise 3.3 Q9 (Part I p. 67). The general skew instance from Part I p. 64 again, now to be halved both ways. Verified: the half-sum is the three by three zero matrix and the half-difference is the matrix itself. This is the sharpest item in the exercise — a matrix that is already skew has no symmetric part at all, and the general recipe still works, returning zero for one half.
• Exercise 3.3 Q10 (Part I pp. 67–68). Four matrices to be split. Verified: (i) three and three above three and minus one, plus zero and two above minus two and zero. (ii) is already symmetric, so its symmetric part is itself and its skew part is the zero matrix — the mirror of Q9 and the second reason to keep the recipe rather than eyeballing. (iii) gives a symmetric part of three, one half and minus five halves; one half, minus two and minus two; minus five halves, minus two and two, and a skew part of zero, five halves and three halves; minus five halves, zero and three; minus three halves, minus three and zero. (iv) gives one and two above two and two, plus zero and three above minus three and zero.
• Exercise 3.3 Q11 and Q12 (Part I p. 68). Verified: Q11, for two symmetric matrices the difference of the two products is skew symmetric, so the first option is right — and the reason is precisely the reversal law of the previous topic. Q12 gives the angle as a third of a half-turn, because the sum of the matrix with its transpose is twice the cosine on the diagonal and zero elsewhere, so the cosine must be one half.
• Miscellaneous Example 24 (Part I pp. 70–71). For two symmetric matrices, the product is symmetric exactly when the two commute. Verified both directions from the reversal law. This is the topic's best link back to the second module: the whole result is one application of the transpose reversal law, and the non-commutativity that looked like a nuisance there is the exact obstruction here.
• Miscellaneous Exercise Q1, Q2 and Q10 (Part I pp. 72–73). Verified: Q1, the difference of the two products of two symmetric matrices is skew symmetric. Q2, sandwiching a matrix between a transpose and the matrix itself preserves whichever of the two kinds it had. Q10, a matrix satisfying both conditions must equal both its transpose and the negative of its transpose, hence its own negative, hence the zero matrix — so the second option is right. Q10 is section 11 and it is the cleanest one-line argument in the chapter.

Figures to have open

• A reflection grid for sections 2 and 3: one square array, the leading diagonal drawn, and every off-diagonal pair joinable across it — first with equal signs, then with opposite ones. Build it with the repo's DataTable component and reuse the diagonal-reflection device from the previous topic.
• A three-line derivation panel for section 4, showing the index substitution, the self-negation and the conclusion. This is the chapter's own argument; the panel is the explanation's staging of it.
• A five-stage strip for section 9 covering build, check, build, check, add back, using Example 22's own matrices.
• A failed-split panel for section 8: the same two brackets without the halves, adding to twice the original. An added device — the chapter states the halves and does not show what happens without them.
• No redraw of a textbook figure is possible or needed: this chapter prints no figure at all. Confirmed on the page image of every one of the forty-two pages.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 3 "Matrices", §3.6, Definitions 4 and 5 and the diagonal derivation, Part I p. 63
• The skew instance in three letters, Part I p. 64
• Theorem 1 and its proof, Part I p. 64
• Theorem 2 and its proof, Part I pp. 64–65
• Example 22, Part I pp. 65–66
• Exercise 3.3, questions 7 to 12, Part I pp. 67–68
• Miscellaneous Example 24, Part I pp. 70–71
• Miscellaneous Exercise on Chapter 3, questions 1, 2 and 10, Part I pp. 72–73
• Summary, the symmetric, skew symmetric and splitting bullets, Part I p. 74