Exercise 3.1 — Matrices

1. 3.1.13 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   In the matrix A=[[2,5,19,-7],[35,-2,5/2,12],[sqrt3,1,-5,17]], write: (i) the order of the matrix, (ii) the number of elements, (iii) the elements a13, a21, a33, a24, a23.

   Hint. Count rows and columns for the order; the number of elements is rows times columns; read each a_ij as row i, column j.

   The matrix has 3 rows and 4 columns, so its order is 3x4. The number of elements is 3x4=12. Reading off entries by row and column: a13 (row 1, col 3) = 19; a21 (row 2, col 1) = 35; a33 (row 3, col 3) = -5; a24 (row 2, col 4) = 12; a23 (row 2, col 3) = 5/2.

   ✦ Order 3x4; 12 elements; a13=19, a21=35, a33=-5, a24=12, a23=5/2.

2. 3.1.23 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

   Hint. List every pair of positive integers whose product is 24 (and separately, 13).

   For 24 elements, list every factor pair (m,n) with mn=24: 1x24, 24x1, 2x12, 12x2, 3x8, 8x3, 4x6, 6x4. For 13 elements, since 13 is prime, the only factor pairs are 1x13 and 13x1.

   ✦ 24 elements: 1x24, 24x1, 2x12, 12x2, 3x8, 8x3, 4x6, 6x4. 13 elements: 1x13, 13x1.

3. 3.1.33 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

   Hint. List every factor pair of 18, then note 5 is prime.

   For 18 elements: 1x18, 18x1, 2x9, 9x2, 3x6, 6x3. For 5 elements, since 5 is prime, the only factor pairs are 1x5 and 5x1.

   ✦ 18 elements: 1x18, 18x1, 2x9, 9x2, 3x6, 6x3. 5 elements: 1x5, 5x1.

4. 3.1.46 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   Construct a 2x2 matrix, A=[a_ij], whose elements are given by: (i) a_ij=(i+j)^2/2 (ii) a_ij=i/j (iii) a_ij=(i+2j)^2/2.

   Hint. Substitute i=1,2 and j=1,2 into each formula to get all four entries.

   (i) a11=(1+1)^2/2=2, a12=(1+2)^2/2=4.5, a21=(2+1)^2/2=4.5, a22=(2+2)^2/2=8. So A=[[2,4.5],[4.5,8]]. (ii) a11=1/1=1, a12=1/2, a21=2/1=2, a22=2/2=1. So A=[[1,0.5],[2,1]]. (iii) a11=(1+2)^2/2=4.5, a12=(1+4)^2/2=12.5, a21=(2+2)^2/2=8, a22=(2+4)^2/2=18. So A=[[4.5,12.5],[8,18]].

   ✦ (i) [[2,4.5],[4.5,8]] (ii) [[1,0.5],[2,1]] (iii) [[4.5,12.5],[8,18]]

5. 3.1.55 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   Construct a 3x4 matrix, whose elements are given by: (i) a_ij=(1/2)|-3i+j| (ii) a_ij=2i-j.

   Hint. Substitute i=1,2,3 and j=1,2,3,4 into each formula, row by row.

   Since each entry only depends on its own row and column index, substitute i and j directly into the formula for every position. (i) Row i=1: (1/2)|-3+j| for j=1..4: 1, 0.5, 0, 0.5. Row i=2: (1/2)|-6+j|: 2.5, 2, 1.5, 1. Row i=3: (1/2)|-9+j|: 4, 3.5, 3, 2.5. (ii) Row i=1: 2(1)-j for j=1..4: 1, 0, -1, -2. Row i=2: 2(2)-j: 3, 2, 1, 0. Row i=3: 2(3)-j: 5, 4, 3, 2.

   ✦ (i) [[1,0.5,0,0.5],[2.5,2,1.5,1],[4,3.5,3,2.5]] (ii) [[1,0,-1,-2],[3,2,1,0],[5,4,3,2]]

6. 3.1.66 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   Find the values of x, y and z from the following equations: (i) [[4,3],[x,5]]=[[y,z],[1,5]] (ii) [[x+y,2],[5+z,xy]]=[[6,2],[5,8]] (iii) [[x+y+z],[x+z],[y+z]]=[[9],[5],[7]].

   Hint. Match corresponding entries; for (ii) solve x+y=6 and xy=8 as a pair; for (iii) use elimination between the three equations.

   (i) Matching entries: y=4, z=3, x=1. (ii) x+y=6, 5+z=5 so z=0, xy=8. Solving x+y=6 and xy=8: x,y are roots of t^2-6t+8=0, giving t=2 or t=4, so x=2,y=4 (or x=4,y=2). (iii) x+y+z=9, x+z=5, y+z=7. Subtracting the second from the first: y=4. Subtracting y+z=7 from x+y+z=9 with y=4: x+z=5 (consistent), and from y+z=7, z=7-4=3. Then x=5-z=5-3=2.

   ✦ (i) x=1,y=4,z=3 (ii) x=2,y=4,z=0 (iii) x=2,y=4,z=3

7. 3.1.74 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   Find the value of a, b, c and d from the equation: [[a-b,2a+c],[2a-b,3c+d]]=[[-1,5],[0,13]].

   Hint. Match all four entries into a system, then eliminate b between the first and third equations.

   Matching entries gives four equations: a-b=-1, 2a+c=5, 2a-b=0, 3c+d=13. Since both the first and third equations involve only a and b, subtracting the first from the third eliminates b: (2a-b)-(a-b)=0-(-1), giving a=1. Then b=a+1=2. From 2a+c=5: c=5-2=3. From 3c+d=13: d=13-9=4.

   ✦ a=1, b=2, c=3, d=4

8. 3.1.81 markNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   A=[a_ij]_(m x n) is a square matrix, if (A) m<n (B) m>n (C) m=n (D) None of these.

   Hint. Recall the definition of a square matrix.

   A matrix is square exactly when its number of rows equals its number of columns, i.e. m=n.

   ✦ (C) m=n

9. 3.1.93 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

   Which of the given values of x and y make the following pair of matrices equal: [[3x+7,5],[y+1,2-3x]], [[0,y-2],[8,4]]? (A) x=-1/3, y=7 (B) Not possible to find (C) y=7, x=-2/3 (D) x=-1/3, y=-2/3.

   Hint. Match all four entries into separate equations for x, and check whether they agree with each other.

   Matching entries gives four equations: 3x+7=0, 5=y-2, y+1=8, 2-3x=4. From 3x+7=0: x=-7/3. From 2-3x=4: x=-2/3. These two values of x contradict each other, so no single x satisfies both equations simultaneously — the matrices can never be equal for any choice of x and y.

   ✦ (B) Not possible to find

10. 3.1.102 marksNCERT Class 12 Mathematics, Matrices, Reprint 2026-27

    The number of all possible matrices of order 3x3 with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512.

    Hint. Count the total number of entries in a 3x3 matrix, and how many independent choices each entry has.

    A 3x3 matrix has 9 entries, and each entry independently can be 0 or 1 (2 choices). So the total count is 2^9=512.

    ✦ (D) 512

Solutions written by the tuition.in editorial team and checked against the NCERT Class 12 Mathematics textbook, Reprint 2026-27 (lemh103.pdf) — Exercise 3.1 (10 questions), Exercise 3.2 (22 questions), Exercise 3.3 (12 questions), Exercise 3.4 (1 question), plus the chapter's Miscellaneous Exercise (11 questions), 56 questions total. Exercise pages were rendered as 300dpi images throughout to read matrix bracket notation accurately, since raw text extraction badly garbled every matrix in the chapter. The old stub taught 'finding the inverse using elementary row/column operations' as a core section, but that method has been fully removed from the current edition — Section 3.7 only proves existence and uniqueness of an inverse, and Exercise 3.4 has shrunk from what older editions had down to a single leftover MCQ, confirmed by reading straight through the section with nothing about row-reducing a matrix anywhere in it. The old stub also collapsed all four real exercises into one invented 30-question group with no solutions file behind it. Every matrix computation in this file (all 56 questions) was independently verified with a Python script using numpy for numeric matrices and sympy for the symbolic ones (F(x)F(y)=F(x+y), the A'A=I system, and the two symbolic quadratic-in-x miscellaneous questions), catching zero discrepancies against the hand-derived working.. Questions are referenced from the NCERT textbook for identification.