Matrices

Transpose of a matrix is a fundamental concept in Class 12 Mathematics and is directly connected with several important topics in the chapter. Questions involving \(A^T\), symmetric matrices, skew-symmetric matrices, and properties of matrix operations frequently test whether a student can correctly interchange rows and columns.

• Accuracy: It prevents the common mistake of assuming that every square matrix remains unchanged after transposition.
• Symmetric matrices: The condition \(A^T=A\) is central to identifying symmetric matrices.
• Skew-symmetric matrices: The condition \(A^T=-A\) is based directly on the transpose operation.
• Matrix algebra: Transpose properties are extensively used in later problems involving products and scalar multiples of matrices.
• Short-answer questions: Direct computation of \(A^T\) is a common source of easy marks when row-column interchange is performed carefully.

Significance for Competitive Entrance Examinations

For competitive examinations, transpose questions are often used as building blocks for more advanced matrix identities. A strong command of the operation helps in solving questions involving matrix equations, symmetric and skew-symmetric decomposition, determinants, and matrix products.

• Remember that \((A^T)^T=A\).
• Remember that \((AB)^T=B^TA^T\); the order of multiplication reverses.
• Remember that \((kA)^T=kA^T\), where \(k\) is a scalar.
• For a symmetric matrix, \(A^T=A\).
• For a skew-symmetric matrix, \(A^T=-A\).
• Always verify the order of the resulting matrix before finalising an answer.

This question is important because it tests a fundamental property of transpose that is repeatedly used in the Matrix chapter. In board examinations, a verification question requires both sides of the identity to be calculated correctly and compared explicitly.

• Stepwise verification: Students should calculate the left-hand side and right-hand side separately before concluding that the identity is verified.
• Arithmetic accuracy: Sign errors, especially subtraction of negative numbers, can invalidate an otherwise correct solution.
• Transpose accuracy: Rows must be converted into columns without changing the values of the entries.
• Presentation: Writing the intermediate matrices makes the verification transparent and helps secure method marks.
• Conceptual foundation: These properties are used later with scalar multiplication, matrix products, symmetric matrices, and skew-symmetric matrices.

• Given transpose: If \(A^T\) is given, obtain \(A\) by taking the transpose again.
• Order awareness: A \(3\times2\) matrix \(A^T\) implies that \(A\) is \(2\times3\).
• Entry tracking: Every element must be placed in its correct position while taking the transpose.
• Verification method: For an identity, calculate the left-hand side and right-hand side separately.
• Arithmetic discipline: A single incorrect matrix entry can affect every subsequent step.

For board examinations, writing the intermediate matrices clearly is advisable because it demonstrates the complete method rather than merely stating the final identity.

Significance for Competitive Entrance Examinations

Transpose properties are frequently embedded in objective and algebraic matrix problems. The ability to recognise the identity immediately can reduce lengthy calculations.

The following results are especially important:

Among these, the product identity deserves particular attention because the order of the factors reverses:

Recognising such identities is useful for simplifying matrix expressions quickly in competitive examinations.

This problem is important because it combines three basic matrix operations: recovering a matrix from its transpose, scalar multiplication, and matrix addition. It also demonstrates how transpose properties can shorten calculations.

• Students must know how to obtain \(A\) when \(A^T\) is given.
• Every entry of a matrix is multiplied by the scalar when finding \(2B\).
• Transpose must be taken only after the matrix operation is completed if using the direct method.
• Alternatively, the identity
  \[ (A+2B)^T=A^T+2B^T \]
  provides a faster route.
• The problem is a good example of how matrix identities can reduce computational work in an examination.

Significance for Competitive Entrance Examinations

For competitive examinations, the most efficient approach is often to apply matrix identities directly instead of calculating unnecessary intermediate matrices.

Here, the expression can immediately be transformed as

This avoids calculating \(A\) and \(A+2B\) separately.

The important identities to remember are:

These identities are particularly useful in objective questions where recognising the correct transformation is faster than performing full matrix calculations.

This is an important question because it tests one of the most fundamental properties of matrix multiplication and transpose. It also tests whether the student understands matrix dimensions before multiplying matrices.

• Dimension check: Always check the orders of matrices before multiplication.
• Product order: If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then \(AB\) is \(m\times p\).
• Transpose of product:
  \[ (AB)^T=B^TA^T. \]
• Order reversal: The matrices appear in reverse order after transposition.
• Matrix multiplication is not entrywise multiplication: A column matrix multiplied by a row matrix produces an outer-product matrix, not a scalar.
• Presentation: Writing the orders and intermediate matrices makes the verification logically complete.

Significance for Competitive Entrance Examinations

The identity

is extremely important in competitive mathematics. It is often used to simplify complicated matrix expressions without performing full matrix multiplication.

Students should distinguish carefully between the following:

Thus, transposition reverses the order of factors.

Another important point is that matrix multiplication is generally not commutative:

when both products exist.

In this question, the distinction is especially clear:

whereas

This makes the order of multiplication an essential part of the solution.

This question is important for the Class 12 board examination because it combines three fundamental ideas: transpose of a matrix, matrix multiplication, and the trigonometric identity \(\sin^2\alpha+\cos^2\alpha=1\).

• It provides a standard method for proving that a matrix is orthogonal.
• Students should be able to calculate \(A^T\) accurately by interchanging rows and columns.
• Every entry of \(A^TA\) should be shown clearly rather than jumping directly to the identity matrix.
• The diagonal terms reduce to \(1\) using
  \[ \sin^2\alpha+\cos^2\alpha=1. \]
• The off-diagonal terms cancel because
  \[ \sin\alpha\cos\alpha-\cos\alpha\sin\alpha=0. \]
• Showing these simplifications explicitly makes the proof complete and earns method marks.

Significance for Competitive Entrance Examinations

Orthogonal matrices occur frequently in objective questions involving matrix properties, inverses, determinants and transformations.

The key recognition is

Therefore, once a matrix is recognised as orthogonal, its inverse can be written immediately without using the adjoint or determinant formula.

For example, if

then immediately

This can substantially reduce calculation time in entrance examinations.

This question is a direct application of the definitions of symmetric and skew-symmetric matrices and is highly relevant for board examinations.

• Remember the two defining conditions:
  \[ A^T=A \]
  for symmetric matrices and
  \[ A^T=-A \]
  for skew-symmetric matrices.
• For a symmetric matrix, corresponding entries across the principal diagonal are equal.
• For a skew-symmetric matrix, corresponding entries across the principal diagonal are negatives of each other.
• The diagonal entries of every skew-symmetric matrix are zero.
• Writing the complete transpose is the safest method in a proof-based board question.
• For method marks, explicitly state the defining condition before comparing the matrices.

Significance for Competitive Entrance Examinations

Symmetric and skew-symmetric matrices are frequently used as building blocks in objective questions and matrix decomposition problems.

An important result is that every square matrix \(A\) can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix:

Here,

is symmetric, while

is skew-symmetric.

This decomposition is useful in higher-level matrix problems and competitive examinations.

For quick identification, check the principal diagonal first. If a matrix is claimed to be skew-symmetric but has a non-zero diagonal element, the claim is immediately false.

This is an important conceptual question because it tests whether the student can apply the definitions of symmetric and skew-symmetric matrices using transpose properties.

• Remember that
  \[ A^T=A \]
  characterises a symmetric matrix.
• Remember that
  \[ A^T=-A \]
  characterises a skew-symmetric matrix.
• For \(A+A^T\), explicitly calculate the matrix and verify that its transpose is the same matrix.
• For \(A-A^T\), verify that its transpose is its negative.
• In a board examination, showing the intermediate matrix calculation prevents loss of method marks.
• The general identities are worth remembering because they can solve similar questions quickly.

Significance for Competitive Entrance Examinations

This concept is especially useful in objective questions where the matrix itself may be complicated but its transpose is easy to identify.

For any square matrix \(A\), immediately recognise:

This allows many multiple-choice questions to be answered without performing complete matrix addition.

The decomposition

is also a frequently useful result in higher-level problems involving symmetric and skew-symmetric matrices.

This question is important because it tests the decomposition of a matrix into its symmetric and skew-symmetric components, a standard concept in NCERT Class 12 Mathematics.

• Students should remember
  \[ \boxed{\frac{1}{2}(A+A^T)} \]
  as the symmetric part of \(A\).
• Students should remember
  \[ \boxed{\frac{1}{2}(A-A^T)} \]
  as the skew-symmetric part of \(A\).
• If \(A\) is already skew-symmetric, its symmetric part is automatically the zero matrix.
• If \(A\) is already symmetric, its skew-symmetric part is automatically the zero matrix.
• Writing every matrix subtraction explicitly avoids sign errors, especially in expressions such as \(a-(-a)\).
• The final decomposition should satisfy
  \[ A=\frac12(A+A^T)+\frac12(A-A^T). \]

Significance for Competitive Entrance Examinations

This problem contains a useful shortcut for objective examinations. Before performing lengthy calculations, inspect the matrix for symmetry or skew-symmetry.

Here,

Therefore, immediately,

and

Thus, the answer can be obtained almost instantly once the structural property of the matrix is recognised.

The decomposition formula is also useful in problems asking for the symmetric or skew-symmetric component of a complicated matrix:

This question is a direct application of the fundamental decomposition theorem for square matrices. It is important for board examinations because students are expected to perform transpose, matrix addition and subtraction, scalar multiplication, and identify symmetry properties accurately.

• Memorise the decomposition:
  \[ \boxed{A=\frac12(A+A^T)+\frac12(A-A^T)}. \]
• Always calculate \(A^T\) carefully before proceeding.
• The first component is always symmetric.
• The second component is always skew-symmetric.
• If the given matrix is already symmetric, the skew-symmetric component is \(O\).
• If the given matrix is already skew-symmetric, the symmetric component is \(O\).
• For full board marks, do not merely state the two matrices; show how they are obtained and, where appropriate, verify their symmetry properties.

Significance for Competitive Entrance Examinations

For competitive examinations, the key advantage is that the decomposition can be obtained mechanically from two standard expressions:

Several shortcuts follow immediately:

• If \(A^T=A\), then \(S=A\) and \(K=O\).
• If \(A^T=-A\), then \(S=O\) and \(K=A\).
• The diagonal entries of every skew-symmetric matrix are zero.
• The diagonal entries of the symmetric component are exactly the diagonal entries of the original matrix.
• For an \(n\times n\) matrix, the symmetric and skew-symmetric parts have complementary off-diagonal structures.

Recognising these patterns can substantially reduce calculation time in objective questions.

This is a high-value conceptual question because it combines three important matrix properties:

1. Transpose of a sum or difference:
   \[(A-B)^T=A^T-B^T.\]
2. Transpose of a product:
   \[(AB)^T=B^TA^T.\]
3. Definition of a skew-symmetric matrix:
   \[A^T=-A\]

For competitive entrance examinations, the fastest approach is to take the transpose immediately and use \(A^T=A\) and \(B^T=B\). This avoids unnecessary numerical calculations.

This question tests several fundamental skills from the chapter: finding the transpose, adding matrices, identifying the identity matrix, and comparing corresponding entries of two matrices.

• Remember that \(A^T\) is obtained by interchanging rows and columns.
• When two matrices are equal, their corresponding entries are equal.
• For an identity matrix of order \(2\),
  \[ I= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}. \]
• Writing the matrix comparison step explicitly helps secure method marks in a descriptive examination.

Significance for Competitive Entrance Examinations

This is an excellent quick-response problem. The cancellation of the off-diagonal terms can be recognised immediately:

Thus, only the diagonal condition remains:

This reduces the matrix problem to a standard trigonometric value in just a few steps.