Matrices - CBSE Class 12 Maths NCERT Solutions

What is a Matrix? Definition and Types

A matrix is a rectangular arrangement of numbers or functions, called elements or entries, arranged in rows and columns. It is typically enclosed in large brackets or parentheses. The order of a matrix is defined by the number of rows (m) and the number of columns (n) it has, denoted as m x n. For example, a matrix with 3 rows and 2 columns has an order of 3x2.

Let's look at various types of matrices:

1. Column Matrix: A matrix having only one column. Its order is m x 1.
2. Row Matrix: A matrix having only one row. Its order is 1 x n.
3. Square Matrix: A matrix in which the number of rows is equal to the number of columns (m = n). Its order is m x m or n x n.
4. Diagonal Matrix: A square matrix in which all non-diagonal elements are zero. The diagonal elements can be anything.
5. Scalar Matrix: A diagonal matrix where all the diagonal elements are equal.
6. Identity Matrix (Unit Matrix): A scalar matrix where all the diagonal elements are 1. It is denoted by 'I'. For a 3x3 identity matrix, it looks like:

\[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

1. Zero Matrix (Null Matrix): A matrix in which all elements are zero. It is denoted by 'O'.

Equality of Matrices: Two matrices A and B are said to be equal if and only if:

1. They are of the same order.
2. Each element of A is equal to the corresponding element of B. That is, a$_{ij}$ = b$_{ij}$ for all i and j.

Operations on Matrices: Addition, Subtraction, and Multiplication

Just like numbers, matrices can be added, subtracted, and multiplied, but with specific rules. Understanding these operations is key to working with matrices.

1. Addition and Subtraction of Matrices

Condition: Two matrices A and B can be added or subtracted if and only if they are of the same order. If A = [a$_{ij}$] and B = [b$_{ij}$] are two matrices of the same order m x n, then:

• Addition (A + B): The sum A + B is a matrix C = [c$_{ij}$] of order m x n, where c$_{ij}$ = a$_{ij}$ + b$_{ij}$ for all i and j. In simple terms, you add corresponding elements.
• Subtraction (A - B): The difference A - B is defined as A + (-1)B. This means you subtract corresponding elements.

Example: If A = $\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix}$ and B = $\begin{pmatrix} 1 & 5 \\ 4 & -2 \end{pmatrix}$, then A + B = $\begin{pmatrix} 2+1 & 3+5 \\ 1+4 & 0-2 \end{pmatrix}$ = $\begin{pmatrix} 3 & 8 \\ 5 & -2 \end{pmatrix}$.

2. Scalar Multiplication of a Matrix

If A = [a$_{ij}$] is a matrix of order m x n and 'k' is a scalar (any real number), then the scalar multiplication kA is a matrix obtained by multiplying each element of A by k. So, kA = [ka$_{ij}$].

Example: If A = $\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix}$ and k = 3, then 3A = $\begin{pmatrix} 3 \times 2 & 3 \times 3 \\ 3 \times 1 & 3 \times 0 \end{pmatrix}$ = $\begin{pmatrix} 6 & 9 \\ 3 & 0 \end{pmatrix}$.

3. Matrix Multiplication

This is the most complex, yet crucial, operation. Pay close attention!

Condition: Two matrices A and B can be multiplied to form the product AB if and only if the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then their product AB will be an m x p matrix.

Process: To find the element c$_{ij}$ of the product matrix C = AB, we take the i-th row of A and the j-th column of B, multiply their corresponding elements, and then sum these products.

Example: Let A = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and B = $\begin{pmatrix} e & f \\ g & h \end{pmatrix}$.
AB = $\begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}$

Key Point: Matrix multiplication is generally not commutative, i.e., AB ≠ BA. Also, it's possible for AB to be defined while BA is not, or vice-versa, or both are defined but not equal. This distinct property is vital to remember.

Properties of Matrix Operations

Understanding the properties of matrix operations is crucial for simplifying expressions and solving matrix equations efficiently. While some properties are similar to those of real numbers, others, especially concerning multiplication, are quite unique.

Properties of Matrix Addition:

1. Commutative Law: If A and B are matrices of the same order, then A + B = B + A.
2. Associative Law: If A, B, and C are matrices of the same order, then (A + B) + C = A + (B + C).
3. Existence of Additive Identity: For any matrix A, there exists a zero matrix O (of the same order as A) such that A + O = O + A = A.
4. Existence of Additive Inverse: For any matrix A, there exists a matrix -A (of the same order) such that A + (-A) = (-A) + A = O, where -A is obtained by multiplying each element of A by -1.

Properties of Scalar Multiplication:

1. k(A + B) = kA + kB, where k is a scalar.
2. (k + l)A = kA + lA, where k and l are scalars.
3. k(lA) = (kl)A.

Properties of Matrix Multiplication:

1. Associative Law: If A, B, and C are matrices such that (AB)C is defined, then A(BC) is also defined, and (AB)C = A(BC).
2. Distributive Law: For matrices A, B, and C, if the operations are defined:

• A(B + C) = AB + AC (Left Distributive Law)
• (A + B)C = AC + BC (Right Distributive Law)

1. Existence of Multiplicative Identity: For every square matrix A of order n, there exists an identity matrix I of order n such that AI = IA = A. The identity matrix plays the role of '1' in matrix multiplication.

Important Non-Property: As mentioned earlier, matrix multiplication is generally NOT commutative. That is, for two matrices A and B, AB is usually not equal to BA. In some special cases, they might be equal, but it's not a general rule. Always keep this in mind when performing matrix multiplication.

Worked Examples on Matrix Operations

• Example 1: Addition and Scalar Multiplication Given matrices A = $\begin{pmatrix} 2 & 4 \\ 3 & 2 \end{pmatrix}$ and B = $\begin{pmatrix} 1 & 3 \\ -2 & 5 \end{pmatrix}$. Find: (i) A + B (ii) 3A - B Solution: (i) A + B: Since A and B are both 2x2 matrices, addition is possible. Step 1: Add corresponding elements. A + B = $\begin{pmatrix} 2+1 & 4+3 \\ 3+(-2) & 2+5 \end{pmatrix}$ Step 2: Perform the additions. A + B = $\begin{pmatrix} 3 & 7 \\ 1 & 7 \end{pmatrix}$ (ii) 3A - B: Step 1: First, calculate 3A by multiplying each element of A by 3. 3A = $\begin{pmatrix} 3 \times 2 & 3 \times 4 \\ 3 \times 3 & 3 \times 2 \end{pmatrix}$ = $\begin{pmatrix} 6 & 12 \\ 9 & 6 \end{pmatrix}$ Step 2: Now, subtract B from 3A (element-wise). 3A - B = $\begin{pmatrix} 6-1 & 12-3 \\ 9-(-2) & 6-5 \end{pmatrix}$ Step 3: Perform the subtractions. 3A - B = $\begin{pmatrix} 5 & 9 \\ 11 & 1 \end{pmatrix}$
• Example 2: Matrix Multiplication (2x2 by 2x2) Given matrices A = $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Find AB. Solution: Step 1: Check compatibility. A is 2x2, B is 2x2. Number of columns in A (2) equals number of rows in B (2). So, multiplication is possible, and the resulting matrix AB will be 2x2. Step 2: Calculate the elements of the product matrix C = AB. C$_{11}$ (1st row of A x 1st col of B) = (1)(5) + (2)(7) = 5 + 14 = 19 C$_{12}$ (1st row of A x 2nd col of B) = (1)(6) + (2)(8) = 6 + 16 = 22 C$_{21}$ (2nd row of A x 1st col of B) = (3)(5) + (4)(7) = 15 + 28 = 43 C$_{22}$ (2nd row of A x 2nd col of B) = (3)(6) + (4)(8) = 18 + 32 = 50 Step 3: Form the resulting matrix. AB = $\begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$
• Example 3: Matrix Multiplication (2x3 by 3x2) Given A = $\begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \end{pmatrix}$ and B = $\begin{pmatrix} 1 & 2 \\ 3 & 0 \\ 1 & 1 \end{pmatrix}$. Find AB. Solution: Step 1: Check compatibility. A is 2x3, B is 3x2. Number of columns in A (3) equals number of rows in B (3). So, multiplication is possible, and the resulting matrix AB will be 2x2. Step 2: Calculate the elements of the product matrix C = AB. C$_{11}$ = (1st row of A x 1st col of B) = (1)(1) + (0)(3) + (2)(1) = 1 + 0 + 2 = 3 C$_{12}$ = (1st row of A x 2nd col of B) = (1)(2) + (0)(0) + (2)(1) = 2 + 0 + 2 = 4 C$_{21}$ = (2nd row of A x 1st col of B) = (2)(1) + (1)(3) + (3)(1) = 2 + 3 + 3 = 8 C$_{22}$ = (2nd row of A x 2nd col of B) = (2)(2) + (1)(0) + (3)(1) = 4 + 0 + 3 = 7 Step 3: Form the resulting matrix. AB = $\begin{pmatrix} 3 & 4 \\ 8 & 7 \end{pmatrix}$

YoLearn.ai Exam Tips for Matrices

To ace your exams on matrices, keep these crucial points in mind:

1. Check Compatibility: Always verify the order of matrices before performing any operation. For addition/subtraction, matrices must be of the same order. For multiplication (AB), the number of columns in A must equal the number of rows in B.
2. Matrix Multiplication Order: Remember that matrix multiplication is generally not commutative (AB ≠ BA). If the question asks for both AB and BA, calculate both separately. Also, if A is m x n and B is n x p, then AB is m x p. This helps verify your result's order.
3. Element-wise Operations: Addition, subtraction, and scalar multiplication are done element-wise. Don't mix this up with matrix multiplication where rows and columns interact.
4. Careful with Signs: When performing subtraction or dealing with negative scalars, pay close attention to signs to avoid calculation errors. A common mistake is $a - (-b)$ becoming $a - b$ instead of $a + b$.
5. Identity Matrix Properties: Understand the role of the identity matrix (I). Remember that for any square matrix A, AI = IA = A. This is frequently used in solving matrix equations.

Practice Questions with Solutions

• Q: Given A = $\begin{pmatrix} 0 & 1 \\ 2 & 3 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix}$. Find 2A + B. A: Step 1: Calculate 2A. 2A = $\begin{pmatrix} 2 \times 0 & 2 \times 1 \\ 2 \times 2 & 2 \times 3 \end{pmatrix}$ = $\begin{pmatrix} 0 & 2 \\ 4 & 6 \end{pmatrix}$ Step 2: Add B to 2A. 2A + B = $\begin{pmatrix} 0+4 & 2+5 \\ 4+6 & 6+7 \end{pmatrix}$ Step 3: Perform the additions. Final answer: $\begin{pmatrix} 4 & 7 \\ 10 & 13 \end{pmatrix}$
• Q: If A = $\begin{pmatrix} -1 & 2 \\ 3 & -4 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & 0 \\ -2 & 1 \end{pmatrix}$, find AB. A: Step 1: Check compatibility. A is 2x2, B is 2x2. Product AB will be 2x2. Step 2: Calculate the elements of AB. (AB)$_{11}$ = (-1)(5) + (2)(-2) = -5 - 4 = -9 (AB)$_{12}$ = (-1)(0) + (2)(1) = 0 + 2 = 2 (AB)$_{21}$ = (3)(5) + (-4)(-2) = 15 + 8 = 23 (AB)$_{22}$ = (3)(0) + (-4)(1) = 0 - 4 = -4 Final answer: $\begin{pmatrix} -9 & 2 \\ 23 & -4 \end{pmatrix}$
• Q: Find values of x, y, z if $\begin{pmatrix} x+y & 2 \\ 5+z & xy \end{pmatrix}$ = $\begin{pmatrix} 6 & 2 \\ 8 & 8 \end{pmatrix}$. A: Step 1: Use the condition for equality of matrices: corresponding elements must be equal. Equation 1: x + y = 6 Equation 2: 5 + z = 8 Equation 3: xy = 8 Step 2: Solve Equation 2 for z. z = 8 - 5 = 3 Step 3: Solve Equation 1 and Equation 3 simultaneously for x and y. From Equation 1, y = 6 - x. Substitute this into Equation 3. x(6 - x) = 8 6x - x$^2$ = 8 x$^2$ - 6x + 8 = 0 (x - 2)(x - 4) = 0 So, x = 2 or x = 4. Step 4: Find corresponding y values. If x = 2, y = 6 - 2 = 4. If x = 4, y = 6 - 4 = 2. Final answer: (x=2, y=4, z=3) or (x=4, y=2, z=3).
• Q: If A = $\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 1 & 0 \\ -2 & 5 \end{pmatrix}$. Check if AB = BA. A: Step 1: Calculate AB. (AB)$_{11}$ = (2)(1) + (-1)(-2) = 2 + 2 = 4 (AB)$_{12}$ = (2)(0) + (-1)(5) = 0 - 5 = -5 (AB)$_{21}$ = (3)(1) + (4)(-2) = 3 - 8 = -5 (AB)$_{22}$ = (3)(0) + (4)(5) = 0 + 20 = 20 AB = $\begin{pmatrix} 4 & -5 \\ -5 & 20 \end{pmatrix}$ Step 2: Calculate BA. (BA)$_{11}$ = (1)(2) + (0)(3) = 2 + 0 = 2 (BA)$_{12}$ = (1)(-1) + (0)(4) = -1 + 0 = -1 (BA)$_{21}$ = (-2)(2) + (5)(3) = -4 + 15 = 11 (BA)$_{22}$ = (-2)(-1) + (5)(4) = 2 + 20 = 22 BA = $\begin{pmatrix} 2 & -1 \\ 11 & 22 \end{pmatrix}$ Step 3: Compare AB and BA. Since the corresponding elements are not equal, AB ≠ BA. Final answer: AB ≠ BA, which confirms that matrix multiplication is not commutative.

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