Types of matrices

What are the types of matrices?

In plain words: the order of a matrix and the pattern of its zeros decide its type. Row and column matrices are thin, square matrices have equal sides, and the special square matrices keep their non-zero elements on the diagonal.

Row and column matrices

NCERT page 39: a column matrix has order m × 1 and a row matrix has order 1 × n. For example

is a column matrix of order 4 × 1, and B = [−½, √5, 2, 3] is a row matrix of order 1 × 4. A point (x, y) of the plane is often written as a column matrix of order 2 × 1 (NCERT page 37).

Square matrices and the diagonal

An m × n matrix is square when m = n, and is then called a square matrix of order n (NCERT page 39). NCERT introduces the diagonal for square matrices: in a square matrix A = [a_ij] of order n, the elements a₁₁, a₂₂, …, a_nn form the diagonal of A (NCERT’s note, page 39). More generally, a rectangular matrix also has main-diagonal entries with equal row and column indices, up to min(m, n); the diagonal, scalar and identity matrix types below are square, following NCERT’s definitions.

The diagonal runs from the top left to the bottom right, through the elements whose row number equals their column number. In NCERT’s example

the diagonal elements are 1, 4 and 6.

Diagonal, scalar and identity matrices

A square matrix B = [b_ij] is a diagonal matrix if b_ij = 0 whenever i ≠ j (NCERT page 40). The diagonal elements themselves may be anything, zero included. NCERT’s examples are [4] and

A diagonal matrix is a scalar matrix if its diagonal elements are all equal to one constant k: NCERT’s examples (page 40) are [3], the 2 × 2 matrix with −1 down the diagonal and the 3 × 3 matrix with √3 down the diagonal.

The identity matrix of order n, written I_n or just I, has 1 on the diagonal and 0 elsewhere:

Observe that a scalar matrix is an identity matrix when k = 1.

NCERT Mathematics Part I (Class 12), Section 3.3, page 40

So the types sit inside one another: every identity matrix is scalar, every scalar matrix is diagonal, and every diagonal matrix is square.

Zero matrix

A matrix is said to be zero matrix or null matrix if all its elements are zero.

NCERT Mathematics Part I (Class 12), Section 3.3, page 41

A zero matrix can have any order: [0], the 2 × 2 and 2 × 3 matrices of zeros, and the row [0 0] are all zero matrices. Each is written O, and its order is read from the context (NCERT page 41).

Equality of matrices

NCERT Definition 2 (page 41): two matrices A = [a_ij] and B = [b_ij] are equal if

• they have the same order, and
• a_ij = b_ij for every i and j.

So two matrices with the same elements in different places are not equal; NCERT’s pair with rows 3, 2; 0, 1 and 2, 3; 0, 1 differ in their first row. A 2 × 3 matrix can never equal a 3 × 2 matrix.

An equation between matrices is many ordinary equations at once, one for each position. That is how unknowns are found:

gives x = −1.5, y = 0, z = 2, a = √6, b = 3 and c = 2 (NCERT page 41).

Solved examples

Common mistakes to avoid

Quick revision summary

• Column matrix: order m × 1; row matrix: order 1 × n; square matrix: order n × n.
• The diagonal of a square matrix is a₁₁, a₂₂, …, a_nn.
• Diagonal: a_ij = 0 for i ≠ j. Scalar: diagonal with all diagonal elements equal to k. Identity I: scalar with k = 1. Zero matrix O: every element 0.
• A = B when the orders match and a_ij = b_ij for every i and j; one matrix equation gives one ordinary equation per position.

Questions students ask

Is a unit matrix the same as an identity matrix?

Yes, unit matrix is another widely used name, beyond the NCERT text, for the identity matrix I_n, the square matrix with every diagonal element 1 and every other element 0; NCERT itself uses only the name identity matrix (Section 3.3, page 40). Both names fit the part it plays in multiplication, that of the number 1: A I = I A = A for every square matrix A of the same order, one of the laws for products. A matrix with every element 1, such as the 2 × 2 matrix of four 1s, is not the identity matrix, and because a few references call that all-ones matrix the unit matrix, identity matrix is the safer name.

Why is a scalar matrix called scalar?

A scalar matrix is called scalar because it is a number k, a scalar, times the identity matrix, kI, and multiplying by it does what multiplying by that number does: (kI)A = A(kI) = kA for every square matrix A of its order. For example, the 2 × 2 scalar matrix with 4 down the diagonal turns the matrix with rows 1, 3 and 2, 5 into the one with rows 4, 12 and 8, 20, from either side. So the one matrix kI does the work of the scalar multiple kA for every matrix A it can multiply. Beyond the NCERT text, scalar matrices are the only square matrices that commute with every square matrix of the same order.

What is a triangular matrix?

A triangular matrix, a type beyond the NCERT text, is a square matrix with every element on one side of the diagonal 0: upper triangular when every element below the diagonal is 0 (a_ij = 0 for i > j), lower triangular when every one above it is 0 (a_ij = 0 for i < j). The matrix with rows 2, −1, 7; 0, 3, 5 and 0, 0, −4 is upper triangular, and its mirror image across the diagonal, with rows 2, 0, 0; −1, 3, 0 and 7, 5, −4, is lower triangular. A matrix is both exactly when it is diagonal, so every diagonal matrix, I included, is triangular. Chapter 4 inverts an upper triangular matrix.

What is the principal diagonal of a matrix, and what is its trace?

The principal diagonal of a square matrix, also called its main or leading diagonal (names beyond the NCERT text), is what NCERT calls the diagonal: a₁₁, a₂₂, …, a_nn, from the top left to the bottom right. The other diagonal, which NCERT does not name, is the secondary diagonal, the elements with i + j = n + 1, such as a₁₃, a₂₂ and a₃₁ of a 3 × 3 matrix. The trace, also beyond the NCERT text, is the sum of the principal diagonal elements: the matrix with rows 19, 1, 0; 2, 25, 3 and 6, 8, 15 has trace 19 + 25 + 15 = 59, and I_n has trace n.

What is the square or any power of a diagonal matrix?

Every positive integer power of a diagonal matrix is again a diagonal matrix, with each diagonal element raised to that power: the 3 × 3 diagonal matrix with diagonal 2, −5, 3 has a square with diagonal 4, 25, 9 and a cube with diagonal 8, −125, 27. This works because the product of two diagonal matrices of the same order is diagonal, each diagonal element being the product of the two in that position, which is also why such matrices commute. In particular I² = I, and indeed Iⁿ = I for every natural number n, while a scalar matrix gives (kI)ⁿ = kⁿI. The zeroth power is I. Negative integer powers exist only when every diagonal entry is non-zero; their entries are the corresponding negative powers.

Can a matrix be a row matrix and a column matrix at the same time?

Yes, but only a 1 × 1 matrix [a] is both a row matrix and a column matrix, since it alone has exactly one row and exactly one column. It is also square, diagonal and scalar, as NCERT’s examples [4] and [3] of order 1 show (page 40), and beyond the NCERT text it counts as triangular too; [1] is the identity matrix I₁, and [0] is the zero matrix of order 1. Every other row matrix has at least two columns, so it is not a column matrix and not square.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Find x, y and z in each equation (NCERT Exercise 3.1, Q6 (i) and (iii), page 42).

   $$\text{(i)}\ \begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$$ $$\text{(ii)}\ \begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$$
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   (i) Matching positions: y = 4, z = 3, x = 1. (ii) Subtracting x + z = 5 from x + y + z = 9 gives y = 4; then z = 7 − 4 = 3 and x = 5 − 3 = 2.

2. Q2. Find a, b, c and d (NCERT Exercise 3.1, Q7, page 42).

   $$\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$$
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   a − b = −1 and 2a − b = 0 give a = 1, b = 2. Then 2 + c = 5 gives c = 3, and 9 + d = 13 gives d = 4.

3. Q3. A = [a_ij]_{m × n} is a square matrix if (A) m < n; (B) m > n; (C) m = n; (D) none of these (NCERT Exercise 3.1, Q8, page 43).

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   A square matrix has as many rows as columns: (C).

4. Q4. Which of these are diagonal, scalar or identity matrices?

   $$\text{(i)}\ \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$$ $$\text{(ii)}\ \begin{bmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$ $$\text{(iii)}\ \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$$
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   (i) Diagonal and scalar (k = 7), not identity. (ii) Diagonal, not scalar (the diagonal elements 2, 0, −1 differ). (iii) None: a₂₁ = 1 is off the diagonal.