IIT Madras
Mathematics 2
Matrices

Matrices

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Question 1: Matrix Basics

Problem Suppose $A = \begin{bmatrix} 2 & 4 & 5 & 1 \ 11 & -2 & 9 & -6 \ -3 & 4 & 7 & 7 \end{bmatrix}$. Which of the following is true about the matrix A? [Hint: The (i, j)-th entry is the entry which is at the i-th row and j-th column.]

Options

  • It is a $4 \times 3$ matrix.
  • It is a $3 \times 4$ matrix.
  • (2, 3)-th entry of the matrix A is 4.
  • (2, 3)-th entry of the matrix A is 9.

Correct Options

  • It is a $3 \times 4$ matrix.
  • (2, 3)-th entry of the matrix A is 9.

Concepts Explained ๐Ÿ’ก

  • Order of a Matrix: The order (or dimension) of a matrix is given by (number of rows) ร— (number of columns).
  • Matrix Entry: An entry is identified by its position, (row, column). The (i, j)-th entry is the element located at the i-th row and j-th column.

Step-by-Step Solution

  1. Determine the Order: The matrix A has 3 horizontal rows and 4 vertical columns. Therefore, its order is $3 \times 4$.
  2. Identify the (2, 3)-th Entry: We need the element in the 2nd row and the 3rd column.
    • Row 2 is [11, -2, 9, -6].
    • The 3rd element in this row is 9.
    • Therefore, the (2, 3)-th entry is 9.

Question 2: Types of Matrices

Problem Which of the following statements is (are) TRUE? [Hint: Recall, the definitions of scalar matrix, diagonal matrix, and identity matrix.]

Options

  • Any diagonal matrix is a scalar matrix.
  • Scalar matrices may not be square matrices.
  • Scalar matrices must be square matrices.
  • Any scalar matrix is an identity matrix.

Correct Option

  • Scalar matrices must be square matrices.

Concepts Explained ๐Ÿ’ก

  • Square Matrix: A matrix with an equal number of rows and columns.
  • Diagonal Matrix: A square matrix where all elements outside the main diagonal are zero. The diagonal elements can be any value. Example: $\begin{bmatrix} 1 & 0 \ 0 & 5 \end{bmatrix}$.
  • Scalar Matrix: A special type of diagonal matrix where all the elements on the main diagonal are equal. Example: $\begin{bmatrix} 7 & 0 \ 0 & 7 \end{bmatrix}$.
  • Identity Matrix (I): A special type of scalar matrix where all the elements on the main diagonal are 1. Example: $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$.

The hierarchy is: Identity Matrix $\subset$ Scalar Matrix $\subset$ Diagonal Matrix $\subset$ Square Matrix.

Analysis of Options

  • Any diagonal matrix is a scalar matrix: False. A diagonal matrix can have different values on its diagonal (e.g., $\begin{bmatrix} 1 & 0 \ 0 & 5 \end{bmatrix}$), while a scalar matrix must have equal values.
  • Scalar matrices may not be square matrices: False. A scalar matrix is a type of diagonal matrix, which, by definition, must be square.
  • Scalar matrices must be square matrices: True. This follows directly from the definition.
  • Any scalar matrix is an identity matrix: False. A scalar matrix can have any value on its diagonal (e.g., $\begin{bmatrix} 7 & 0 \ 0 & 7 \end{bmatrix}$), while the identity matrix must have 1s.

Question 3: Matrix Representation of Linear Equations

Problem Given the system of linear equations: $7x_1 + 10x_2 + 12x_3 = 36$ $8x_1 + 4x_2 - 9x_3 = 11$ $4x_1 - x_2 + 3x_3 = 10$ if the matrix representation is $Ax = b$, where $x = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}$, then what are A and b?

Correct Option

  • $A = \begin{bmatrix} 7 & 10 & 12 \ 8 & 4 & -9 \ 4 & -1 & 3 \end{bmatrix}, b = \begin{bmatrix} 36 \ 11 \ 10 \end{bmatrix}$

Concepts Explained ๐Ÿ’ก

A system of linear equations can be written in the compact matrix form $Ax = b$.

  • A is the coefficient matrix, containing the coefficients of the variables in each equation. Each row in the matrix corresponds to an equation.
  • x is the variable vector, a column vector containing the variables.
  • b is the constant vector, a column vector containing the constants from the right-hand side of each equation.

Step-by-Step Solution

  1. Form the Coefficient Matrix (A): Write the coefficients of $x_1, x_2, x_3$ from each equation as a row in the matrix.
    • From Eq 1: [7, 10, 12]
    • From Eq 2: [8, 4, -9]
    • From Eq 3: [4, -1, 3] (Note that $-x_2$ has a coefficient of -1).
  2. Form the Constant Vector (b): Write the constants from the right-hand side of each equation as a column vector.
    • [36, 11, 10] This matches the first option.

Question 4: Rules for Matrix Operations

Problem Which of the following statements is (are) TRUE?

Options

  • Addition of two matrices is possible only if the number of columns in the first matrix is same as the number of rows in the second matrix.
  • Addition of two matrices is possible only if the orders of both the matrices are the same.
  • Defining $AB$ is possible if the number of rows in the matrix $A$ is same as the number of columns in the matrix $B$.
  • Defining $AB$ is possible if the number of columns in the matrix $A$ is same as the number of rows in the matrix $B$.

Correct Options

  • Addition of two matrices is possible only if the orders of both the matrices are the same.
  • Defining $AB$ is possible if the number of columns in the matrix $A$ is same as the number of rows in the matrix $B$.

Concepts Explained ๐Ÿ’ก

  • Matrix Addition: To add two matrices, they must have the exact same dimensions (e.g., both are $3 \times 2$). The addition is then performed element-wise.
  • Matrix Multiplication: For the product $AB$ to be defined, the “inner” dimensions must match. If A is an $m \times \mathbf{n}$ matrix and B is an $\mathbf{n} \times p$ matrix, the product is possible, and the resulting matrix will have the order $m \times p$.

Question 5: Matrix Operations and Orders

Problem Suppose $P = \begin{bmatrix} 3 & -1 & 7 \ 4 & 0 & 1 \ 2 & -5 & 2 \end{bmatrix}, Q = \begin{bmatrix} 1 & 4 & -9 \end{bmatrix}, R = \begin{bmatrix} 0 & -3 & 10 \end{bmatrix}, D = \begin{bmatrix} -2 \ 4 \ 5 \end{bmatrix}$. Which of the following statements are true?

Correct Options

  • The matrix $PD$ is of order $3 \times 1$.
  • The matrix $QD$ is of order $1 \times 1$.
  • The matrix $DQ$ is of order $3 \times 3$.
  • The product $QR$ is not defined.
  • The addition $P + Q$ is not defined.
  • The addition $P + D$ is not defined.

Concepts Explained ๐Ÿ’ก

  • Order of Product: If A is ($m \times n$) and B is ($n \times p$), then AB is ($m \times p$).
  • Condition for Addition: Matrices must have the same order to be added.

Step-by-Step Analysis

  1. List the orders:
    • P: $3 \times 3$
    • Q: $1 \times 3$
    • R: $1 \times 3$
    • D: $3 \times 1$
  2. Check each statement:
    • PD: P($3 \times \mathbf{3}$) and D($\mathbf{3} \times 1$). Inner dimensions (3 and 3) match. The resulting order is $3 \times 1$.
    • QD: Q($1 \times \mathbf{3}$) and D($\mathbf{3} \times 1$). Inner dimensions (3 and 3) match. The resulting order is $1 \times 1$.
    • DQ: D($3 \times \mathbf{1}$) and Q($\mathbf{1} \times 3$). Inner dimensions (1 and 1) match. The resulting order is $3 \times 3$.
    • QR: Q($1 \times 3$) and R($1 \times 3$). Inner dimensions (3 and 1) do not match. Product is not defined.
    • P + Q: P($3 \times 3$) and Q($1 \times 3$). Orders are different. Addition is not defined.
    • P + D: P($3 \times 3$) and D($3 \times 1$). Orders are different. Addition is not defined.

Question 6: Matrix Properties and Equations

Problem Choose the set of correct options.

Correct Options

  • If $A$ and $B$ are square matrices of order 3 and $A + B = 0$, then $B = -A$.
  • If $A$ is a scalar matrix of order 3, $B$ is a non-zero square matrix of order 3 and $AB = 0$, then $A = 0$.

Concepts Explained ๐Ÿ’ก

  • Additive Inverse: For any matrix A, there exists a matrix -A such that $A + (-A) = 0$. The matrix $B$ such that $A+B=0$ is the additive inverse of A.
  • Zero Divisors: In matrix multiplication, it’s possible for the product of two non-zero matrices to be the zero matrix ($AB=0$ where $A\neq0, B\neq0$).
  • Scalar Matrix Property: A scalar matrix is of the form $kI$. Multiplying by a scalar matrix is equivalent to multiplying by the scalar: $(kI)B = k(IB) = kB$.

Analysis of Options

  • If $A^2 = I$, then $A = I$ or $A = -I$: False. This is not generally true for matrices. A counterexample is the reflection matrix $A = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$. Here, $A^2 = I$, but $A$ is neither $I$ nor $-I$.
  • If $A^2 = 0$, then $A = 0$: False. A matrix where $A^k=0$ for some $k$ is called nilpotent. A counterexample is $A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}$. Here, $A^2 = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$, but $A$ is not the zero matrix.
  • If $A + B = 0$, then $B = -A$: True. This is the definition of the additive inverse in matrix algebra. Subtracting A from both sides gives $B = 0 - A = -A$.
  • If A is a scalar matrix… and $AB = 0$, then $A = 0$: True. Let A be the scalar matrix $kI$. The equation becomes $(kI)B = 0$, which simplifies to $kB = 0$. Since we are given that B is a non-zero matrix, the only way for the product $kB$ to be the zero matrix is if the scalar $k=0$. If $k=0$, then the matrix $A=kI$ is the zero matrix.

Question 7: Column-wise Matrix Multiplication

Problem If $A$ is a square matrix of order 2 whose first column is denoted by $C_1$ and second column is denoted by $C_2$ and $B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}$, then choose the set of correct options.

Correct Options

  • The first column of $AB$ is $b_{11}C_1 + b_{21}C_2$.
  • The second column of $AB$ is $b_{12}C_1 + b_{22}C_2$.

Concepts Explained ๐Ÿ’ก

Matrix multiplication can be viewed column by column. The j-th column of a product $AB$ is the matrix $A$ multiplied by the j-th column of the matrix $B$. This product results in a linear combination of the columns of $A$, with the coefficients coming from the corresponding column of $B$.

Step-by-Step Solution

Let $A = [C_1 | C_2]$ (representing A by its columns).

  • First Column of AB: This is calculated as $A \times (\text{first column of B})$. $A \begin{bmatrix} b_{11} \ b_{21} \end{bmatrix} = [C_1 | C_2] \begin{bmatrix} b_{11} \ b_{21} \end{bmatrix} = b_{11}C_1 + b_{21}C_2$.
  • Second Column of AB: This is calculated as $A \times (\text{second column of B})$. $A \begin{bmatrix} b_{12} \ b_{22} \end{bmatrix} = [C_1 | C_2] \begin{bmatrix} b_{12} \ b_{22} \end{bmatrix} = b_{12}C_1 + b_{22}C_2$.

Question 8: Existence of Matrices

Problem Choose the set of correct options.

Correct Options

  • There exist some real matrices $A$ and $B$, such that $AB = BA$.
  • There exists some real $2 \times 2$ matrix $A$, such that $A^2 + A + I = 0$.

Concepts Explained ๐Ÿ’ก

  • Commutativity: Matrix multiplication is not generally commutative ($AB \neq BA$). However, there are many specific cases where it is.
  • Idempotent Matrix: A matrix such that $A^2 = A$.
  • Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation. This theorem can be used to prove the existence of matrices that satisfy certain polynomial equations.

Analysis of Options

  • There exist some real matrices A and B, such that AB = BA: True. Many examples exist. For instance, if $A=I$ (the identity matrix), then $IB = B$ and $BI = B$, so $IB=BI$. Another example is if $B$ is the inverse of $A$ ($B=A^{-1}$), then $AA^{-1} = A^{-1}A = I$.
  • There do not exist any…: False, as shown above.
  • There does not exist any real matrix A, such that Aยฒ = A: False. Both the zero matrix ($0^2=0$) and the identity matrix ($I^2=I$) are examples. They are called idempotent matrices.
  • There exists some real 2x2 matrix A, such that Aยฒ + A + I = 0: True. By the Cayley-Hamilton theorem, a matrix must satisfy its characteristic polynomial. The polynomial $\lambda^2 + \lambda + 1 = 0$ has complex roots. A real matrix can have complex eigenvalues (as long as they appear in conjugate pairs). A rotation matrix for an angle of $120^\circ$ (or $2\pi/3$ radians) is an example of such a matrix.

Question 9: Scalar Matrix Operations

Problem Suppose A is a $3 \times 3$ scalar matrix and (1,1)-th entry of the matrix A is 4. Suppose B is a $3 \times 3$ square matrix such that (i, j)-th entry is equal to $i^2 + j^2$. Find the (2,2)-th entry of the matrix $2A + B$.

Answer: 16

Concepts Explained ๐Ÿ’ก

  • Scalar Matrix: A diagonal matrix where all diagonal entries are equal. All non-diagonal entries are 0.
  • Matrix Operations: Scalar multiplication (kA) and addition (A+B) are performed element-wise. The (i,j)-th entry of kA+B is k \cdot a_{ij} + b_{ij}.

Step-by-Step Solution

  1. Determine the (2,2)-th entry of A: A is a scalar matrix, and its (1,1)-th entry is 4. This means all its diagonal entries are 4, and all non-diagonal entries are 0. So, the (2,2)-th entry of A, aโ‚‚โ‚‚, is 4.
  2. Determine the (2,2)-th entry of B: The formula for an entry in B is bแตขโฑผ = iยฒ + jยฒ. For the (2,2)-th entry, i=2 and j=2. So, bโ‚‚โ‚‚ = 2ยฒ + 2ยฒ = 4 + 4 = 8.
  3. Calculate the (2,2)-th entry of 2A + B: The (2,2)-th entry of the final matrix is 2 \cdot aโ‚‚โ‚‚ + bโ‚‚โ‚‚. 2 * (4) + 8 = 8 + 8 = 16.

Question 10: Cayley-Hamilton Theorem

Problem Suppose $A = \begin{bmatrix} -1 & 2 \ -4 & 7 \end{bmatrix}$ and $A^2 - \alpha A + I = 0$ for some $\alpha \in \mathbb{R}$. Find the value of $\alpha$.

Answer: 6

Concepts Explained ๐Ÿ’ก

Cayley-Hamilton Theorem: This fundamental theorem states that any square matrix satisfies its own characteristic equation.

  • Characteristic Equation: For a matrix A, the characteristic equation is given by det(A - ฮปI) = 0, where ฮป is a variable.

Step-by-Step Solution

  1. Find the characteristic equation of A: First, find the matrix A - ฮปI: $A - \lambda I = \begin{bmatrix} -1 & 2 \ -4 & 7 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1-\lambda & 2 \ -4 & 7-\lambda \end{bmatrix}$
  2. Calculate the determinant: det(A - ฮปI) = (-1-\lambda)(7-\lambda) - (2)(-4) = -7 + \lambda - 7\lambda + \lambda^2 + 8 = \lambda^2 - 6\lambda + 1 So, the characteristic equation is $\lambda^2 - 6\lambda + 1 = 0$.
  3. Apply the Cayley-Hamilton Theorem: The theorem states that A satisfies this equation. We replace ฮป with A (and the constant term 1 with 1 \cdot I): $A^2 - 6A + I = 0$
  4. Compare and find ฮฑ: We are given the equation $A^2 - \alpha A + I = 0$. Comparing this with the equation from the theorem, we can see that $\alpha = 6$.

Determinants (part 2)Practice Assignment 1