Exercise 3.1 answers: Matrices

Open NCERT p. 42Matches NCERT’s answer

(i) The order of the matrix

1. Count the rows: there are 3.
2. Count the columns: there are 4.
3. The order is always written rows × columns.

Answer3 × 4

(ii) The number of elements

1. The number of elements is rows × columns.
2. That is 3 × 4.

Answer12

(iii) Write the elements a13, a21, a33, a24, a23

1. a_ij means the entry in row i, column j.
2. a₁₃ is row 1, column 3, which is 19.
3. a₂₁ is row 2, column 1, which is 35.
4. a₃₃ is row 3, column 3, which is −5.
5. a₂₄ is row 2, column 4, which is 12.
6. a₂₃ is row 2, column 3, which is 5/2.

Answera₁₃ = 19, a₂₁ = 35, a₃₃ = −5, a₂₄ = 12, a₂₃ = 5/2

Watch this explained “The double subscript”, 8:29 into An ordered rectangular array, and what its order tells you before anything else

Open NCERT p. 42Matches NCERT’s answer

1. A matrix of order m × n has m × n elements.
2. So the order must be a pair of counting numbers whose product is the given number of elements.
3. For 24: list every pair of factors of 24, both ways round, since 2 × 12 and 12 × 2 are different orders.
4. 24 = 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1 — eight orders.
5. 13 is a prime number, so its only factor pairs are 1×13 and 13×1 — two orders.

AnswerFor 24 elements: 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1. For 13 elements: 1×13, 13×1.

Watch this explained “Counting backwards”, 11:44 into An ordered rectangular array, and what its order tells you before anything else

Open NCERT p. 42Matches NCERT’s answer

1. A matrix of order m × n has m × n elements, so the order must be a factor pair of the given count.
2. For 18: list every factor pair both ways round.
3. 18 = 1×18, 2×9, 3×6, 6×3, 9×2, 18×1 — six orders.
4. 5 is prime, so its only factor pairs are 1×5 and 5×1 — two orders.

AnswerFor 18 elements: 1×18, 2×9, 3×6, 6×3, 9×2, 18×1. For 5 elements: 1×5, 5×1.

Watch this explained “Counting backwards”, 11:44 into An ordered rectangular array, and what its order tells you before anything else

Open NCERT p. 42Matches NCERT’s answer

(i) aij = (i + j)²⁄2

1. A 2×2 matrix has rows i = 1, 2 and columns j = 1, 2.
2. Put each pair (i, j) into the rule (i+j)²÷2.
3. a₁₁ = (1+1)²÷2 = 2, a₁₂ = (1+2)²÷2 = 9/2.
4. a₂₁ = (2+1)²÷2 = 9/2, a₂₂ = (2+2)²÷2 = 8.

Answer[[2, 9/2], [9/2, 8]]

(ii) aij = i⁄j

1. Put each pair (i, j) into the rule i÷j.
2. a₁₁ = 1÷1 = 1, a₁₂ = 1÷2 = 1/2.
3. a₂₁ = 2÷1 = 2, a₂₂ = 2÷2 = 1.

Answer[[1, 1/2], [2, 1]]

(iii) aij = (i + 2j)²⁄2

1. Put each pair (i, j) into the rule (i+2j)²÷2.
2. a₁₁ = (1+2)²÷2 = 9/2, a₁₂ = (1+4)²÷2 = 25/2.
3. a₂₁ = (2+2)²÷2 = 8, a₂₂ = (2+4)²÷2 = 18.

Answer[[9/2, 25/2], [8, 18]]

Watch this explained “Building from a rule”, 13:34 into An ordered rectangular array, and what its order tells you before anything else

Open NCERT p. 42Matches NCERT’s answer

(i) aij = 1⁄2 |−3i + j|

1. Rows go i = 1, 2, 3 and columns go j = 1, 2, 3, 4.
2. Work out −3i + j for each position, then take its size (ignore the minus sign), then halve it.
3. Row 1 (i=1): −3+j gives −2,−1,0,1, so sizes are 2,1,0,1, halved: 1, 1/2, 0, 1/2.
4. Row 2 (i=2): −6+j gives −5,−4,−3,−2, so sizes are 5,4,3,2, halved: 5/2, 2, 3/2, 1.
5. Row 3 (i=3): −9+j gives −8,−7,−6,−5, so sizes are 8,7,6,5, halved: 4, 7/2, 3, 5/2.

Answer[[1, 1/2, 0, 1/2], [5/2, 2, 3/2, 1], [4, 7/2, 3, 5/2]]

(ii) aij = 2i − j

1. Work out 2i − j for each row and column position.
2. Row 1 (i=1): 2−j gives 1, 0, −1, −2.
3. Row 2 (i=2): 4−j gives 3, 2, 1, 0.
4. Row 3 (i=3): 6−j gives 5, 4, 3, 2.

Answer[[1, 0, −1, −2], [3, 2, 1, 0], [5, 4, 3, 2]]

Watch this explained “Building from a rule”, 13:34 into An ordered rectangular array, and what its order tells you before anything else

Open NCERT p. 42Matches NCERT’s answer

(i) 4 3 ; x 5 = y z ; 1 5

1. Two matrices are equal only when the entries in matching positions are equal.
2. Top-left: 4 = y, so y = 4.
3. Top-right: 3 = z, so z = 3.
4. Bottom-left: x = 1.

Answerx = 1, y = 4, z = 3

(ii) x+y 2 ; 5+z xy = 6 2 ; 5 8

1. Match each position: x+y = 6, 5+z = 5, xy = 8.
2. From 5+z = 5, z = 0.
3. x+y = 6 and xy = 8 means x and y are two numbers that add to 6 and multiply to 8.
4. Those numbers are 2 and 4 (in either order).

Answer{x, y} = {2, 4}, z = 0

(iii) x+y+z ; x+z ; y+z = 9 ; 5 ; 7

1. Match each row: x+y+z = 9, x+z = 5, y+z = 7.
2. Subtract the second equation from the first: y = 4.
3. Put y = 4 into the third equation: 4+z = 7, so z = 3.
4. Put z = 3 into the second equation: x+3 = 5, so x = 2.

Answerx = 2, y = 4, z = 3

Watch this explained “More than one answer”, 16:04 into Equality as two demands, and why the orders have to agree before the entries are looked at

Open NCERT p. 42Matches NCERT’s answer

1. Match each position of the two matrices to get four equations: a−b = −1, 2a+c = 5, 2a−b = 0, 3c+d = 13.
2. Subtract the first equation from the third: (2a−b) − (a−b) = 0 − (−1), which gives a = 1.
3. Put a = 1 into a−b = −1: 1−b = −1, so b = 2.
4. Put a = 1 into 2a+c = 5: 2+c = 5, so c = 3.
5. Put c = 3 into 3c+d = 13: 9+d = 13, so d = 4.

Answera = 1, b = 2, c = 3, d = 4

Watch this explained “When they do not”, 14:16 into Equality as two demands, and why the orders have to agree before the entries are looked at

Open NCERT p. 43Matches NCERT’s answer

1. A square matrix is one where the number of rows equals the number of columns.
2. Here m is the number of rows and n is the number of columns.
3. So the matrix is square exactly when m = n.

Answer(C) m = n

Watch this explained “Square”, 3:37 into The named shapes — row, column, square, diagonal, scalar, identity, zero

Open NCERT p. 43Matches NCERT’s answer

1. Match each position: 3x+7 = 0, 5 = y−2, y+1 = 8, 2−3x = 4.
2. From 3x+7 = 0, x = −7/3.
3. From 2−3x = 4, x = −2/3.
4. These two values of x do not agree, so no single value of x can satisfy both positions.
5. Since x cannot be found consistently, the matrices cannot be made equal for any x and y.

Answer(B) Not possible to find

Watch this explained “No answer at all”, 17:48 into Equality as two demands, and why the orders have to agree before the entries are looked at

Open NCERT p. 43Matches NCERT’s answer

1. A 3×3 matrix has 9 positions to fill.
2. Each position can independently be 0 or 1, so there are 2 choices per position.
3. The total count is 2 multiplied by itself 9 times, that is 2⁹.
4. 2⁹ = 512.

Answer(D) 512

Watch this explained “Two questions”, 15:51 into The named shapes — row, column, square, diagonal, scalar, identity, zero