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CBSE · Class 12 · Mathematics

Addition and Scalar Multiplication of Matrices (Ch 3: 3.4.1–3.4.4)

2 concepts · 16 practice questions with answers · Free to read

Addition and subtraction of matrices

From daily life

Two shops record the number of pens and books sold each day. Adding the two tables entry by entry gives the total for both shops.

Simple definition

Matrices of the same order are added (or subtracted) entry by entry. Addition is commutative and associative; O is the additive identity; the matrix −A (every entry negated) satisfies A + (−A) = O.

Formula: A + B = [aᵢⱼ + bᵢⱼ], A + B = B + A, A + O = A

Rule 1: Add or subtract — Work position by position.

  • Easy: [1 2 ; 3 4] + [5 6 ; 7 8] = [6 8 ; 10 12].
  • Medium: [3 −1 2] + [0 4 −5] = [3 3 −3].
  • Exam-type: [2 5 ; −1 3] − [4 1 ; −2 6] = [−2 4 ; 1 −3].

Rule 2: Find the unknown matrix — Move terms across like numbers.

  • Easy: If A + X = B with A = [1 2] and B = [4 7], then X = B − A = [3 5].
  • Medium: X + [2 1 ; 0 3] = [5 4 ; 1 3] gives X = [3 3 ; 1 0].
  • Exam-type: [1 2 3] + [4 5] is not defined because the orders differ.
Remember

Only matrices of the same order can be added or subtracted.

Common mistake

Trying to add a 2 × 3 matrix to a 3 × 2 matrix.

Memory clue

Same size, then add seat by seat.

Practice questions

Easy
  1. 1Find [1 3 ; 2 4] + [2 0 ; 1 5].
  2. 2Find [5 2] − [1 6].
  3. 3Is [1 2 3] + [1 2] defined?
Medium
  1. 4Find X if X + [1 2 ; 3 4] = [6 5 ; 4 9].
  2. 5Find A + B and B + A for A = [2 −1 ; 0 3] and B = [1 4 ; −2 0].
  3. 6Find −A for A = [3 −2 ; 0 5].
Challenging
  1. 7Find A + (−A) for A = [4 1 ; −3 2].
  2. 8Find X if X − [2 0 ; 1 1] = [3 5 ; 2 2].
Show answers (try them yourself first)
  1. 1[3 3 ; 3 9]
  2. 2[4 −4]
  3. 3No, the orders 1 × 3 and 1 × 2 differ
  4. 4X = [5 3 ; 1 5]
  5. 5Both give [3 3 ; −2 3], so A + B = B + A
  6. 6[−3 2 ; 0 −5]
  7. 7The zero matrix [0 0 ; 0 0]
  8. 8X = [5 5 ; 3 3]

Scalar multiplication

From daily life

A recipe table lists the cups of each ingredient. Making double the recipe means multiplying every number in the table by 2.

Simple definition

The product kA of a number k and a matrix A multiplies every entry of A by k. Then k(A + B) = kA + kB and (k + l)A = kA + lA.

Formula: kA = [k aᵢⱼ], k(A + B) = kA + kB

Rule 1: Multiply by a scalar — Multiply every entry.

  • Easy: 3 [1 2 ; 0 4] = [3 6 ; 0 12].
  • Medium: −2 [1 −3 5] = [−2 6 −10].
  • Exam-type: ½ [4 6 ; 8 10] = [2 3 ; 4 5].

Rule 2: Combine scalar multiples — Do the multiplications first, then add.

  • Easy: With A = [1 2 ; 3 4], B = [0 1 ; 1 0]: 2A + 3B = [2 4 ; 6 8] + [0 3 ; 3 0] = [2 7 ; 9 8].
  • Medium: With A = [2 1], B = [1 3]: 3A − 2B = [6 3] − [2 6] = [4 −3].
  • Exam-type: Solve 2X + A = B for A = [1 3 ; 5 7], B = [5 9 ; 11 15]: 2X = [4 6 ; 6 8], X = [2 3 ; 3 4].
Remember

Scalar multiplication touches every entry, including zeros (which stay zero).

Common mistake

Multiplying only the first row or one entry by the scalar.

Memory clue

"Spread the number over every seat."

Practice questions

Easy
  1. 1Find 4 [2 −1 ; 0 3].
  2. 2Find −[3 −5 2].
  3. 3Find ⅓ [3 6 9].
Medium
  1. 4Find 2A + B for A = [1 0 ; 2 3], B = [4 1 ; 0 2].
  2. 5Find 3A − 2B for A = [2 1 ; 0 4], B = [1 0 ; 3 2].
  3. 6Find X if 2X = [6 −4 ; 8 10].
Challenging
  1. 7Find A and B if A + B = [5 3 ; 4 7] and A − B = [1 1 ; 2 3].
  2. 8Find k if kA = [6 12 ; 18 24] for A = [1 2 ; 3 4].
Show answers (try them yourself first)
  1. 1[8 −4 ; 0 12]
  2. 2[−3 5 −2]
  3. 3[1 2 3]
  4. 4[2 0 ; 4 6] + [4 1 ; 0 2] = [6 1 ; 4 8]
  5. 5[6 3 ; 0 12] − [2 0 ; 6 4] = [4 3 ; −6 8]
  6. 6X = [3 −2 ; 4 5]
  7. 7Add: 2A = [6 4 ; 6 10], so A = [3 2 ; 3 5]; then B = [2 1 ; 1 2]
  8. 8k = 6