Matrices Class 12: notes and important questions

Matrices — revision notes

1. Order, construction and types

• An \(m \times n\) matrix has \(m\) rows and \(n\) columns; \(a_{ij}\) sits in row \(i\), column \(j\). A matrix with \(k\) elements can have order \(m \times n\) for every factor pair \(mn = k\).
• Types: row, column, square, diagonal (\(a_{ij} = 0\) for \(i \ne j\)), scalar (diagonal with equal diagonal entries, \(kI\)), identity \(I\), zero matrix \(O\).
• Equal matrices: same order and equal corresponding entries — equate entries to get equations.

2. Operations

• \(A + B\) and \(kA\): entry by entry, same order only. \(A + B = B + A\), \(k(A + B) = kA + kB\).
• \(AB\) is defined when (columns of \(A\)) = (rows of \(B\)); \((m \times n)(n \times p) = m \times p\). Entry \((i, j)\) = row \(i\) of \(A\) · column \(j\) of \(B\).
• Associative and distributive, but not commutative: in general \(AB \ne BA\), so \((A + B)^2 = A^2 + AB + BA + B^2\).
• \(AB = O\) does not force \(A = O\) or \(B = O\) (e.g. \(\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}0&0\\1&0\end{bmatrix} = O\)).

3. Transpose, symmetric and skew-symmetric matrices

• \((A^T)^T = A\), \((A + B)^T = A^T + B^T\), \((kA)^T = kA^T\), \((AB)^T = B^TA^T\) (order reverses).
• Symmetric: \(A^T = A\). Skew-symmetric: \(A^T = -A\) (diagonal entries all \(0\)).
• Every square matrix: \(A = \tfrac12(A + A^T) + \tfrac12(A - A^T)\) = symmetric + skew-symmetric, uniquely.
• To test a product, take its transpose and use \((XY)^T = Y^TX^T\) with \(A^T = \pm A\).

4. Invertible matrices

• \(B\) is the inverse of square \(A\) if \(AB = BA = I\). The inverse is unique: \(B = B(AC) = (BA)C = C\).
• \((AB)^{-1} = B^{-1}A^{-1}\). If \(A\) satisfies \(A^2 + pA + qI = O\) with \(q \ne 0\), then \(A\left(-\tfrac1q(A + pI)\right) = I\).
• If \(A\) is invertible and \(AX = C\), then \(X = A^{-1}C\).

Worked example 1

\(A = \begin{bmatrix}1&3\\0&1\end{bmatrix}\): \(A^2 = \begin{bmatrix}1&6\\0&1\end{bmatrix}\), and in general \(A^n = \begin{bmatrix}1&3n\\0&1\end{bmatrix}\) (each multiplication adds \(3\) to the corner).

Worked example 2

If \(A\) is symmetric, \(B^TAB\) is symmetric: \((B^TAB)^T = B^TA^T(B^T)^T = B^TAB\).

Worked example 3

\(A = \begin{bmatrix}4&-1\\2&1\end{bmatrix}\) satisfies \(A^2 - 5A + 6I = O\) (check: \(A^2 = \begin{bmatrix}14&-5\\10&-1\end{bmatrix}\)). So \(A(5I - A) = 6I\) and \(A^{-1} = \tfrac16\begin{bmatrix}1&1\\-2&4\end{bmatrix}\).

Common errors

• Writing \((AB)^T = A^TB^T\) — the order must reverse.
• Expanding \((A + B)^2\) as \(A^2 + 2AB + B^2\) or cancelling: \(AB = AC\) does not give \(B = C\) unless \(A\) is invertible.
• Multiplying matrices whose orders do not match, or mixing up rows × columns when stating an order.
• Checking only \(AB = I\) when the question asks you to verify an inverse: show \(BA = I\) as well.
• Forgetting that a skew-symmetric matrix has zero diagonal.

Board-exam tips

• Write the order of each product before multiplying — it catches most errors.
• In “show that” questions, show the full matrix after each multiplication; step marks are given for \(A^2\) and for the final combination separately.
• Elementary row operations are not in the current syllabus; use the definition \(AB = BA = I\) or a polynomial identity.