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CBSE Class 12 · Mathematics

Matrices

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Matrices” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Matrix

A matrix is a rectangular array of numbers as taught, written with a row×column size. A 2×3 has 2 rows and 3 columns. A matrix is an array-word, not a determinant (that is the next chapter). Order is m×n, not “big”.

Calling a 2×3 a 3×2 because it “looks tall” is a miss.

Figure. A matrix is a grid with an order. This 2 × 3 array stores six numbers; the highlighted 7 sits at row 2, column 3. The order is counted first — two rows, three columns — before any arithmetic.

How it works

  1. Count rows then columnsThe order.
  2. Write the arrayThe matrix.
  3. Refuse det-talk hereA different chapter.

2Types of Matrices

Types: row, column, square, zero, identity, diagonal as taught. A type is a look-test. I₂ = [[1,0],[0,1]]. Zero matrix has every entry 0. Type is not “important-looking”.

A diagonal with a 2 in the corner off-diagonal is not diagonal.

Figure. Type is the extra mark on the grid: one row, one column, square with ones on the diagonal (identity), or every entry zero. The test is the pattern of entries, not a new kind of number.

How it works

  1. Name the look (row / I / zero / …)The type.
  2. Run it on the arrayIn or out.
  3. Keep I as 1s on the main diagonalThe identity.

3Operations on Matrices

Operations: add same-size entrywise; multiply AB when A’s columns match B’s rows as taught. [[1,2],[3,4]]+[[5,0],[1,1]] = [[6,2],[4,5]]. Multiply is row-by-column, not entrywise (unless Hadamard was separately named).

Adding a 2×2 to a 2×3 is a size-miss.

Figure. AB is defined only when the column count of A equals the row count of B. Here 2 × 3 times 3 × 2 is allowed and the product is 2 × 2. Addition would need the same order on both sides, which these two do not share.

How it works

  1. Match sizes for add; columns-to-rows for multiplyThe gate.
  2. Add entrywise or multiply row×columnThe operation.
  3. Keep AB and BA as possibly differentNot commutative.

Add 2×2

[[1,2],[3,4]] + [[5,0],[1,1]].

  • (1,1)1+5=6
  • (1,2)2+0=2
  • Result[[6,2],[4,5]]

Pro tip. Same order; add matching seats.

4Transpose of a Matrix

Transpose: Aᵀ swaps rows and columns as taught. (Aᵀ)ᵀ=A. [[1,2,3]]ᵀ is a 3×1. Transpose is a flip, not an inverse.

Calling Aᵀ the inverse because both have a superscript is a miss.

Figure. Transpose swaps row index with column index. The marked 0 starts at (1, 2) and lands at (2, 1). Order 2 × 3 becomes 3 × 2; the six numbers are the same set, newly addressed.

How it works

  1. Swap rows with columnsAᵀ.
  2. Keep the entries, only seats moveThe flip.
  3. Refuse inverse-talk as this headingA different type.

5Symmetric and Skew Symmetric Matrices

Symmetric means Aᵀ=A; skew-symmetric means Aᵀ=−A as taught. A square is required. [[1,2],[2,3]] is symmetric. Diagonal of a skew-symmetric is 0 as taught.

A non-square called symmetric is a miss.

Figure. Symmetric means each off-diagonal pair matches, so A equals A transpose — here both off-diagonal entries are 5. Skew-symmetric means each pair is opposite and the diagonal is forced to 0, so A equals minus A transpose.

How it works

  1. Need square, then compare Aᵀ to A or −AThe test.
  2. Keep skew-diagonal 0 if taughtThe extra.
  3. Refuse Aᵀ=A⁻¹ as this headingDifferent words.

6Invertible Matrices

Invertible: a square A has B with AB=BA=I as taught. Inverse is that B, written A⁻¹. Not every square is invertible (det 0 waits for the next chapter). One 2×2 inverse if the lesson computed one.

Writing A⁻¹ as 1/A entrywise is a miss.

Figure. A is invertible when there exists A inverse with AA inverse = I. That partner exists only if det A is not 0. The det-0 box is the closed door, not a second inverse.

How it works

  1. Ask for B with AB=IInvertible.
  2. Keep it squareThe gate.
  3. Refuse entrywise 1/aᵢⱼNot the inverse.

7A definition is a test you can run

Matrix-word is a test: order, a type, add/multiply, transpose, or inverse. If you only say “box”, you have a heading.

A spreadsheet-sticker is not the test.

Figure. Addition is a test on order: same shape, then add entrywise. 2 × 3 plus 2 × 3 is allowed. 2 × 3 plus 3 × 2 fails the test even though both hold six numbers.

How it works

  1. Name order, type, or an operationThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

8Name the given before the unknown

The given is an array and a size. The unknown is the type or the result. Copy 2×3 before you multiply as if it were 3×2.

Using AB when A is 2×3 and B is 2×2 fails the column-row match.

Figure. Write the two given matrices and their orders before you reach for the product. If the inner counts fail the test, there is no AB to find — the unknown stays blank on purpose.

How it works

  1. Copy the ordersThe given.
  2. Name the operation or typeThe unknown.
  3. Then computeGiven first.
[[1,2],[3,4]] + [[5,0],[1,1]] is
  1. [[6,2],[4,5]]
  2. [[5,2],[3,5]]
  3. A 2×3

Entrywise add.

Notes

  • The official chapter title is “Matrices”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • (Aᵀ)ᵀ=A
  • AB needs n_col(A)=n_row(B)
  • invertible: AB=BA=I

Recap

Hold these pegs from the official chapter “Matrices”. The wording is ExamMaster’s teaching, not a textbook recap.

Matrix
A matrix is a rectangular array of numbers as taught, written with a row×column size.
Types of Matrices
Types: row, column, square, zero, identity, diagonal as taught.
Operations on Matrices
Operations: add same-size entrywise; multiply AB when A’s columns match B’s rows as taught.
Transpose of a Matrix
Transpose: Aᵀ swaps rows and columns as taught.
Symmetric and Skew Symmetric Matrices
Symmetric means Aᵀ=A; skew-symmetric means Aᵀ=−A as taught.
Invertible Matrices
Invertible: a square A has B with AB=BA=I as taught.

Practise Matrices

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