Matrix Operations — Addition, multiplication
Matrices
6
JEE Qs
8%
Hard
75
min
Precisely practice matrix multiplication with different orders, focus on the row-by-column method, and internalize its non-commutative nature to avoid common conceptual and calculation errors.
🧮 Key Formulas
✅ Key Points for JEE
- 1Matrix addition and subtraction are defined only for matrices of the same order, and are performed element-wise.
- 2Matrix multiplication (AB) is defined only if the number of columns in A equals the number of rows in B. The resulting matrix AB has the number of rows of A and the number of columns of B (m x n multiplied by n x p gives m x p).
- 3Matrix multiplication is generally NOT commutative (AB ≠ BA). Always check if both products are even defined and if their orders are compatible.
- 4Matrix multiplication is associative: (AB)C = A(BC). It is also distributive over addition: A(B+C) = AB + AC and (A+B)C = AC + BC.
- 5The product of two non-zero matrices can be a zero matrix (unlike real numbers).
⚠️ Common Mistakes
- ✕Multiplying matrices whose orders are incompatible for multiplication, or incorrectly determining the order of the resulting product matrix.
- ✕Assuming that matrix multiplication is commutative (i.e., AB = BA), which is false in general and can lead to incorrect results.
- ✕Errors in calculating elements during matrix multiplication by failing to consistently use the row-by-column method (i.e., dot product of the i-th row of the first matrix with the j-th column of the second matrix for (AB)_ij).
- ✕Confusing scalar multiplication with matrix multiplication or attempting element-wise multiplication for two matrices (which is not standard matrix multiplication).
📝 Practice Questions
See allQ22.Let M denote the set of all real matrices of order 3 × 3 and let S = {−3, −2, −1, 1, 2}. Let S1 = {A = [aij] ∈M : A = AT and aij ∈ S, ∀i, j}, S2 = {A = [aij] ∈M : A = −AT and aij ∈ S, ∀i, j}, S3 = {A = [aij] ∈M : a11 + a22 + a33 = 0 and aij ∈ S, ∀i, j}. If n ( S1 ∪2 US3) = 125α, then α equals _______
Q24.Let S = ∈Z : Am2 + Am = 3I , where A = 2 −1 . Then n(S) is equal to ______. {m −A−6} [ 1 0 ]
Q4. If A, B , and (adj (A−1) + adj (B−1)) are non-singular matrices of same order, then the inverse of A(adj (A−1) + adj (B−1))−1 B , is equal to (1) AB−1 + A−1 B (2) adj (B−1) + adj (A−1) BA−1 (3) AB−1 (4) 1 (adj(B) + adj(A)) |A| + |B| |AB|
Q16.If I = ∫ 0π 3 dx, then ∫210 sin4x sinx+cos4x cos xx 2 2 x sin x+cos (1) π2 (2) π2 12 4 (3) π2 (4) π2 16 8 ∣∣ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2025 (23 Jan Shift 2) JEE Main Previous Year Paper
Q11.Let A = [aij] = [ log5log51288 log4log4255 ] . If Aij is the cofactor of aij, Cij = ∑2k=1 aikAjk, 1 ≤i, j ≤2, and C = [Cij], then 8|C| is equal to : (1) 288 (2) 222 (3) 242 (4) 262
Q5. Let A = [aij] be a matrix of order 3 × 3, with aij = (√2)i+j . If the sum of all the elements in the third row of A2 is α + β√2, α, β ∈Z, then α + β is equal to : (1) 280 (2) 224 (3) 210 (4) 168
NCERT Chapters
- Class 12 Maths Ch 3: Matrices