Matrix and its order
A seat plan of a classroom lists names in rows and columns. Finding "row 3, seat 2" is how we point to an entry of a matrix.
A matrix is a rectangular arrangement of numbers in rows and columns. In this lesson a matrix is written with rows separated by semicolons, so [1 2 3 ; 4 5 6] has two rows and three columns. Its order is rows × columns, here 2 × 3. The entry in row i and column j is aᵢⱼ.
Formula: Order = m × n, number of entries = m × n
Rule 1: Read the order and an entry — Count rows first, then columns.
- Easy: A = [1 2 3 ; 4 5 6] has order 2 × 3 and a₂₃ = 6.
- Medium: B = [2 5 ; 7 1 ; 0 3] has order 3 × 2 and a₃₁ = 0.
- Exam-type: A matrix with 12 entries can have 6 orders: 1×12, 12×1, 2×6, 6×2, 3×4, 4×3.
Rule 2: Build a matrix from a formula — Put each pair (i, j) into the formula.
- Easy: 2 × 2 with aᵢⱼ = i × j is [1 2 ; 2 4].
- Medium: 2 × 3 with aᵢⱼ = 2i − j is [1 0 −1 ; 3 2 1].
- Exam-type: 3 × 3 with aᵢⱼ = (i − j)² is [0 1 4 ; 1 0 1 ; 4 1 0].
The first number of the order counts rows, the second counts columns.
Reading aᵢⱼ as column i, row j; the first index is always the row.
Rows then Columns: "RC cola".
Practice questions
- 1Write the order of [2 5 ; 7 1 ; 0 3].
- 2How many entries does a 4 × 5 matrix have?
- 3For A = [3 8 1 ; 2 6 9] write a₂₃ and a₁₂.
- 4Construct the 2 × 2 matrix with aᵢⱼ = i + 2j.
- 5Construct the 2 × 3 matrix with aᵢⱼ = i² − j.
- 6How many different orders are possible for a matrix with 18 entries?
- 7Write the 3 × 3 matrix with aᵢⱼ = 1 when i = j and 0 otherwise.
- 8How many 2 × 2 matrices can be formed using only the entries 0 and 1?
Show answers (try them yourself first)
- 13 × 2
- 220
- 3a₂₃ = 9 and a₁₂ = 8
- 4[3 5 ; 4 6]
- 5[0 −1 −2 ; 3 2 1]
- 66 orders: 1×18, 18×1, 2×9, 9×2, 3×6, 6×3
- 7The identity matrix I of order 3: [1 0 0 ; 0 1 0 ; 0 0 1]
- 82⁴ = 16
Types of matrices
Some seat plans have a single row, some have a single column, and some are square. Matrices are named in the same way by their shape and by their pattern of zeros.
A row matrix has one row; a column matrix has one column; a square matrix has equal rows and columns. In a square matrix the diagonal runs from top-left to bottom-right. A diagonal matrix has zeros everywhere outside the diagonal; a scalar matrix is a diagonal matrix with equal diagonal entries; an identity matrix I has all diagonal entries 1; a zero matrix O has all entries 0.
Formula: Identity ⊂ scalar ⊂ diagonal ⊂ square
Rule 1: Name the matrix — First check the shape, then the zeros.
- Easy: [4 7 2] is a row matrix and [3 ; 1 ; 5] is a column matrix.
- Medium: [2 0 ; 0 7] is diagonal, and [4 0 ; 0 4] is scalar.
- Exam-type: [1 2 3 ; 0 4 5 ; 0 0 6] is upper triangular (zeros below the diagonal).
Rule 2: Climb the ladder — Every identity matrix is scalar, every scalar matrix is diagonal.
- Easy: I = [1 0 ; 0 1] is also a scalar and a diagonal matrix.
- Medium: [5 0 ; 0 5] is scalar but not identity.
- Exam-type: [1 0 ; 0 2] is diagonal but not scalar.
A zero matrix of any order is allowed; it is the matrix version of the number 0.
Calling [2 0 ; 0 7] a scalar matrix; the diagonal entries must be equal.
Identity is the strictest: diagonal entries are all exactly 1.
Practice questions
- 1Name the type of [5 1 9 2].
- 2Name the type of [2 0 ; 0 7].
- 3Name the type of [4 0 ; 0 4].
- 4Name the type of [2 5 1 ; 0 3 4 ; 0 0 6].
- 5Is every diagonal matrix upper triangular?
- 6Is a zero matrix of order 2 × 2 a diagonal matrix?
- 7Is every scalar matrix diagonal? Is every diagonal matrix scalar?
- 8Name the type of the 3 × 1 matrix [0 ; 0 ; 0].
Show answers (try them yourself first)
- 1Row matrix (1 × 4)
- 2Diagonal matrix, not scalar
- 3Scalar matrix
- 4Upper triangular matrix
- 5Yes, because all entries below the diagonal are zero
- 6Yes, all off-diagonal entries are zero (it is also scalar with k = 0)
- 7Scalar ⇒ diagonal is true; diagonal ⇒ scalar is false, for example [1 0 ; 0 2]
- 8A column matrix, in fact a zero matrix
Equality of matrices
Two seat plans are the same only if they have the same shape and the same name in every seat. Matrices are equal in exactly the same way.
Two matrices are equal if they have the same order and every pair of corresponding entries is equal.
Formula: A = B ⇔ same order and aᵢⱼ = bᵢⱼ for all i, j
Rule 1: Solve for unknowns — Equate matching entries and solve the equations.
- Easy: [x 3 ; 2 y] = [5 3 ; 2 7] gives x = 5, y = 7.
- Medium: [2x + 1 4 ; 3 y − 2] = [7 4 ; 3 5] gives x = 3 and y = 7.
- Exam-type: [x + y 2 ; 5 x − y] = [9 2 ; 5 1] gives x + y = 9 and x − y = 1, so x = 5, y = 4.
Rule 2: Check the order first — Matrices of different orders can never be equal.
- Easy: [1 2 3] and [1 ; 2 ; 3] are not equal (1 × 3 against 3 × 1).
- Medium: [1 2 ; 3 4] and [1 2 ; 3 5] are not equal because a₂₂ differs.
- Exam-type: [x² 4 ; 9 y³] = [25 4 ; 9 8] gives x = 5 or −5 and y = 2.
One difference in one entry is enough to make two matrices unequal.
Equating entries which are not in the same position.
Same order, then same seat, same entry.
Practice questions
- 1Find a and b if [a 2 ; 3 b] = [4 2 ; 3 9].
- 2Find x if [x + 1 5] = [6 5].
- 3Are [1 2 3] and [1 ; 2 ; 3] equal?
- 4Find x and y if [2x − y 4 ; 1 x + y] = [5 4 ; 1 4].
- 5Find x and y if [x² 4 ; 9 y³] = [49 4 ; 9 27].
- 6Find a and b if [2a + b 5 ; 1 a − b] = [8 5 ; 1 1].
- 7Find x, y, z, w if [3x 2y ; z 4w] = [6 8 ; 5 12].
- 8Can [1 2 ; 3 4] equal [1 2 ; 3 5]?
Show answers (try them yourself first)
- 1a = 4, b = 9
- 2x = 5
- 3No, their orders (1 × 3 and 3 × 1) differ
- 42x − y = 5 and x + y = 4 give x = 3, y = 1
- 5x = 7 or −7, and y = 3
- 62a + b = 8 and a − b = 1 give a = 3, b = 2
- 7x = 2, y = 4, z = 5, w = 3
- 8No, the entries in the second row, second column differ (4 and 5)