Matrices PDF

## Matrices A large number of physical phenomenon are governed by linear differential equations which are solved by reducing them into a system of simultaneous linear equations. Matrices originated as mere stores of information but, at present, have found very wide application. They play a very vi...

## Matrices A large number of physical phenomenon are governed by linear differential equations which are solved by reducing them into a system of simultaneous linear equations. Matrices originated as mere stores of information but, at present, have found very wide application. They play a very vital role not only in mathematics but also in communication theory, network analysis, theory of structures, quantum mechanics, biology, sociology, economics, psychology, statistics etc. Matrices play the key role in providing suitable criteria for testing the consistency or otherwise of a system of linear equations. ### 3.1 Definitions A set of mn numbers (real or complex) arranged in a rectangular array having m rows (horizontal lines) and n columns (vertical lines), the numbers being enclosed by brackets [] or (), is called an mxn matrix (read as "m by n matrix"). An mxn matrix is also called a matrix of order mxn. Each of the mn numbers is called an element of the matrix. For example, $$ \begin{bmatrix} 3 & 2 & -15\\ 0 & 4 & 5 \end{bmatrix} $$ is a 2 x 3 matrix or matrix of order 2 x 3. It has two rows and three columns. The numbers 2, 1, 5, 3, 0, 4 are its elements. An m x n matrix is usually written as: $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22} & a_{23} & ... & a_{2n}\\ ... & & & & \\ a_{m1} & a_{m2} & a_{m3} & ... & a_{mn} \end{bmatrix} $$ Here each element has two suffixes. The first suffix indicates the row and the second suffix indicates the column in which the element lies. Thus, a₂₃ is the element lying in the second row and third column, a_ij is the element lying in the ith row and jth column. For brevity, a matrix is usually denoted by a single capital letter A or B or C etc. Thus, an m x n matrix A may be written as: A = [a_ij]_mxn or A = [a_ij], where i = 1, 2, 3, ....., m; j = 1, 2, 3, ....., n. **Real Matrix.** A matrix is said to be real if all its elements are real numbers。 For example, $$ \begin{bmatrix} \sqrt{5} & -3\\ 0 & -\sqrt{27} \end{bmatrix} $$ is a real matrix。 **Square Matrix.** A matrix in which the number of rows is equal to the number of columns is called a square matrix, otherwise, it is said to be a rectangular matrix。 Thus, a matrix A = [a_ij]_mxn is a square matrix if m = n and a rectangular matrix if m≠n. A square matrix having n rows and n columns is called "a square matrix of order n" or "an n-rowed square matrix", e.g., $$ \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix} $$ is a 3-rowed square matrix. The elements a₁₁, a₂₂, a₃₃ of a square matrix are called its diagonal elements and the diagonal along which these elements lie is called the principal diagonal. In a square matrix A = [a_ij], i = j (i) for elements along the principal diagonal, i < j (ii) for elements above the principal diagonal, i > j (iii) for elements below the principal diagonal, i ≠ j (iv) for non-diagonal elements, The sum of the diagonal elements of a square matrix is called its trace. Thus, trace of the n rowed square matrix A = [a_ij] is, a₁₁ + a₂₂ + a₃₃ + .... + a_nn = Σa_ii **Row Matrix.** A matrix having only one row and any number of columns i.e., a matrix of order 1 x n is called a row matrix. e.g., [2 5 -3 0] is a row matrix. **Column Matrix.** A matrix having only one column and any number of rows i.e., a matrix of order m x 1 is called a column matrix. e.g., $$ \begin{bmatrix} \sqrt{2}\\ 0\\ 1\\ \end{bmatrix} $$ is a column matrix. **Null Matrix.** A matrix in which each element is zero is called a null matrix or a zero matrix. A null matrix of order m x n is denoted by O_mxn e.g., $$ \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{bmatrix} $$ = O_3x2, $$ \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{bmatrix} $$ = O_2x4 **Sub-matrix.** A matrix obtained from a given matrix A by deleting some of its rows or column or both is called a sub-matrix of A. Thus, $$ \begin{bmatrix} 3 & 0\\ 1 & 4\\ \end{bmatrix} $$ is a sub-matrix of $$ \begin{bmatrix} 3 & 5 & 0\\ 1 & 5 & 7\\ 0 & - 1 & 2\\ 3 & 7 & 6\\ 1 & 4 & -2\\ \end{bmatrix} $$ obtained by deleting the first row, second and fourth columns of A. **Diagonal Matrix.** A square matrix in which all non-diagonal elements are zero is called a diagonal matrix. Thus, A = [a_ij]_nxn is a diagonal matrix if a_ij = 0 for i ≠ j. For example, $$ \begin{bmatrix} 2 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 0\\ \end{bmatrix} $$ is a diagonal matrix. An n-rowed diagonal matrix is briefly written as diag. [d₁, d₂, ......., d_n], where d₁, d₂, ......, d_n are the diagonal elements. Thus, the above diagonal matrix A can be written as diag. [2, -1, 0]. **Scalar Matrix.** A diagonal matrix in which all the diagonal elements are equal to a scalar, say k, is called a scalar matrix. Thus a scalar matrix is a square matrix in which all non-diagonal elements are zero and all diagonal elements are equal to a scalar, say k. i.e., A =[a_ij]_nxn is a scalar matrix if a_ij = 0 when i ≠ j k when i = j For example, $$ \begin{bmatrix} 2 & 0 & 0\\ 0 & 0 & -5\\ 0 & 0 & 2\\ \end{bmatrix} $$, $$ \begin{bmatrix} -5 & 0 & 0\\ 0 & -5 & 0\\ 0 & 0 & -5\\ \end{bmatrix} $$ are scalar matrices. **Identity Matrix.** A scalar matrix in which each diagonal element is unity (i.e., 1) is called a identity matrix or unit matrix. Thus, a unit matrix is a square matrix in which all non-diagonal elements are zero and all diagonal elements are equal to 1. i.e., A = [a_ij]_nxn is a unit matrix if a_ij = 0 when i ≠ j 1 when i = j A unit matrix of order n is denoted by I_n. If the order is evident, it may be simply denoted by I. Thus, $$ \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix} $$ = I₂, $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ = I₄ **Upper Triangular Matrix.** A square matrix in which all the elements below the principal diagonal are zero is called an upper triangular matrix. Thus, A = [a_ij]_nxn is an upper triangular matrix if a_ij = 0 for i > j. For example, $$ \begin{bmatrix} 2 & 3 & 4\\ 0 -1 & 5\\ 0 & 0 & 3\\ \end{bmatrix} $$ , is an upper triangular matrix. **Lower Triangular Matrix.** A square matrix in which all the elements above the principal diagonal are zero is called a lower triangular matrix. Thus, A = [a_ij]_nxn is a lower triangular matrix if a_ij = 0 for i < j. For example, $$ \begin{bmatrix} -1 & 0 & 0\\ 5 & 6 & 0\\ 3 & 2 & 0\\ \end{bmatrix} $$ is a lower triangular matrix. **Triangular Matrix.** A square matrix in which all the elements either below or above the principal diagonal are zero is called a triangular matrix. Thus, a triangular matrix is either upper triangular or lower triangular. **Equal Matrices.** Two matrices A and B are said to be equal (written as A = B) if and only if they have the same order and their corresponding elements are equal. Thus, if A = [a_ij]_mxn and B = [b_ij]_pxq, then A = B if and only if (i) m = p and n = q (ii) a_ij = b_ij for all i and j. ### 3.2 Addition and Subtraction of Matrices Two matrices are said to be conformable for addition if they have the same order. If A and B are two matrices of the same order, then their sum A + B is a matrix each element of which is obtained by adding the corresponding elements of A and B. In general, if A =[a_ij]_mxn and B = [b_ij]_mxn then A + B = C = [c_ij]_mxn where c_ij = a_ij + b_ij Similarly, if A and B are two matrices of the same order, then their difference A - B is a matrix whose elements are obtained by subtracting the elements of B from the corresponding elements of A. In general, if A =[a_ij]_mxn and B = [b_ij]_mxn then A - B = C = [c_ij]_mxn where c_ij = a_ij - b_ij For example, if $$ \begin{bmatrix} 2 & 5 & 1\\ 3 & 0 & 4 \end{bmatrix} $$ and $$ \begin{bmatrix} 1 & -6 & 2\\ -1 & 5 & 7 \end{bmatrix} $$ then $$ \begin{bmatrix} 2 & 5 & 1\\ 3 & 0 & 4 \end{bmatrix} $$ + $$ \begin{bmatrix} 1 & -6 & 2\\ -1 & 5 & 7 \end{bmatrix} $$= $$ \begin{bmatrix} 3 & -1 & 3\\ 2 & 5 & 11 \end{bmatrix} $$ $$ \begin{bmatrix} 2 & 5 & 1\\ 3 & 0 & 4 \end{bmatrix} $$ - $$ \begin{bmatrix} 1 & -6 & 2\\ -1 & 5 & 7 \end{bmatrix} $$= $$ \begin{bmatrix} 1 & 11 & -1 \\ 4 & -5 & -3 \end{bmatrix} $$ ### 3.3 Multiplication of a Matrix by a Scalar The product of a matrix A = [a_ij] by a scalar k is denoted by kA and is obtained by multiplying every element of A by k. Thus, kA = [ka_ij] If $$ A=\begin{bmatrix} a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3}\\ c_{1} & c_{2} & c_{3} \end{bmatrix} $$ then kA = $$ \begin{bmatrix} ka_{1} & ka_{2} & ka_{3}\\ kb_{1} & kb_{2} & kb_{3}\\ kc_{1} & kc_{2} & kc_{3} \end{bmatrix} $$ In particular, - A = (-1) A = $$ \begin{bmatrix} -a_{1} & -a_{2} & -a_{3}\\ -b_{1} & -b_{2} & -b_{3}\\ \end{bmatrix} $$ ### 3.4 Properties of Matrix Addition (i) Matrix addition is commutative i.e., A + B = B + A. (ii) Matrix addition is associative i.e., (A + B) + C = A + (B + C). (iii) For any matrix A, there exists a null matrix O of the same order as A such that A + O = O + A = A. (iv) For any matrix A, there exists a matrix - A of the same order as A such that A + (-A) = (-A) + A = O. ### Illustrative Examples **Example 1.** Find x, y, z and w if $$3\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -2\\ 6\\ -1 \end{bmatrix} + \begin{bmatrix} 2 & 4 & w\\ x + y & - 1 & 2ω \\ x + 4 & 2ω + 3 & -1 + z + w \end{bmatrix} $$ **Sol.** The given equation is: $$3\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -2 & 4 & w\\ x + y & - 1 & 2ω \\ x + 4 & 2ω + 3 & -1 + z + w \\ \end{bmatrix} $$ Equating the corresponding elements on the two sides 3x = x + 4, 3y = 6 + x + y, 3z = 1 + z + w, 2x = 4, 2y = 6 + x, x = 2, y = 4, z = 1, 2z = -1 + w, w = 3. **Example 2.** Express $$ \begin{bmatrix} 2 & 5 & -7 \\ -9 & 12 & 4 \\ 15 & -13 & 6 \end{bmatrix} $$ as the sum of a lower triangular matrix and an upper triangular matrix with zero leading diagonal. **Sol.** Let $$ L = \begin{bmatrix} a & 0 & 0\\ b & d & 0\\ c & e & f\\ \end{bmatrix} $$ be the lower triangular matrix and $$ U = \begin{bmatrix} 0 & p & q\\ 0 & 0 & r\\ 0 & 0 & 0\\ \end{bmatrix} $$ the upper triangular matrix with zero leading diagonal such that $$ \begin{bmatrix} 2 & 5 & -7 \\ -9 & 12 & 4 \\ 15 & -13 & 6 \end{bmatrix} $$ = $$ \begin{bmatrix} a & 0 & 0 \\ b & d & 0\\ c & e & f \end{bmatrix} $$ + $$ \begin{bmatrix} 0 & p & q\\ 0 & 0 & r\\ 0 & 0 & 0\\ \end{bmatrix} $$ Equating the corresponding elements on the two sides 2 = a, 5 = p, -7= q, -9 = b, 12 = c, 4 = r, 15 = d, 13 = e, 6 = f $$ L = \begin{bmatrix} 2 & 0 & 0 \\ -9 & 12 & 0\\ 15 & -13 & 6 \end{bmatrix} $$ and $$ U = \begin{bmatrix} 0 & 5 & -7 \\ 0 & 0 & 4\\ 0 & 0 & 0\\ \end{bmatrix} $$ ### 3.5. Matrix Multiplication Two matrices A and B are said to be conformable for the product AB (in this very order of A and B) if the number of columns in A (called the pre-factor) is equal to the number of rows in B (called the post-factor). Thus, if the orders of A and B are mx n and p x q respectively, then (i) AB is defined if number of columns in A = number of rows in B, i.e., if n = p. (ii) BA is defined if number of columns in B = number of rows in A, i.e., if q = m. Let A =[a_ij]_mxn and B = [b_ij]_nxp be two matrices conformable for the product AB, then AB is defined as the matrix C = [c_ij]_mxp where c_ij = a₁₁b_1j + a₁₂b_2j + ..... + a_1nb_nj = Σa_ikb_kj (i, j)th element of AB = sum of the products of the elements of ith row of A with the corresponding elements of jth column of B. i.e., The rule for multiplication of two conformable matrices is called row-by-column method. Consider $$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$ and $$ \begin{bmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ b_{31} & b_{32} \end{bmatrix} $$ Orders of A and B are 3 x 3 and 3 x 2 respectively. AB is defined and is of order 3 x 2. $$ \begin{bmatrix} C_{11} & C_{12}\\ C_{21} & C_{22}\\ C_{31} & C_{32} \end{bmatrix} $$ = AB = $$ \begin{bmatrix} C_{11} & C_{12}\\ C_{21} & C_{22}\\ C_{31} & C_{32} \end{bmatrix} $$ where C₁₁ = sum of products of elements of 1st row of A and 1st column of B $$C_{11} = [\begin{bmatrix} a_{11} & a_{12} & a_{13} \end{bmatrix}]\begin{bmatrix} b_{11}\\ b_{21}\\ b_{31}\\ \end{bmatrix} = a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} $$ C₁₂ = sum of products of elements of 1st row of A and 2nd column of B $$C_{12} = [\begin{bmatrix} a_{11} & a_{12} & a_{13} \end{bmatrix}]\begin{bmatrix} b_{12}\\ b_{22}\\ b_{32}\\ \end{bmatrix} = a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} $$ C₂₁ = sum of products of elements of 2nd row of A and 1st column of B $$C_{21} = [\begin{bmatrix} a_{21} & a_{22} & a_{23} \end{bmatrix}]\begin{bmatrix} b_{11}\\ b_{21}\\ b_{31}\\ \end{bmatrix} = a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} $$ C₂₂ = sum of products of elements of 2nd row of A and 2nd column of B $$C_{22} = [\begin{bmatrix} a_{21} & a_{22} & a_{23} \end{bmatrix}]\begin{bmatrix} b_{12}\\ b_{22}\\ b_{32}\\ \end{bmatrix} = a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} $$ C₃₁ = sum of products of elements of 3rd row of A and 1st column of B $$C_{31} = [\begin{bmatrix} a_{31} & a_{32} & a_{33} \end{bmatrix}]\begin{bmatrix} b_{11}\\ b_{21}\\ b_{31}\\ \end{bmatrix} = a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} $$ C₃₂ = sum of products of elements of 3rd row of A and 2nd column of B $$C_{32} = [\begin{bmatrix} a_{31} & a_{32} & a_{33} \end{bmatrix}]\begin{bmatrix} b_{12}\\ b_{22}\\ b_{32}\\ \end{bmatrix} = a_{31}b_{12} + a_{32}b_{22} + a_{33}b_{32} $$ Thus, AB = $$ \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} \\ a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} & a_{31}b_{12} + a_{32}b_{22} + a_{33}b_{32} \end{bmatrix} $$ Note. Another useful notation to remember matrix multiplication. $$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$= $$ \begin{bmatrix} R_{1} \\ R_{2}\\ R_{3}\\ \end{bmatrix} $$ where R₁ = 1st Row. R₂ = 2nd Row R₃ = 3rd Row $$ B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\\ b_{31} & b_{32} \end{bmatrix} $$ = $$ \begin{bmatrix} C_{1} & C_{2} \end{bmatrix} $$ where C₁ = 1st column, C₂ = 2nd column AB= $$ \begin{bmatrix} R_{1} \\ R_{2}\\ R_{3}\\ \end{bmatrix} $$ $$ \begin{bmatrix} C_{1} & C_{2} \end{bmatrix} $$= $$ \begin{bmatrix} R_{1}C_{1} & R_{1}C_{2} \\ R_{2}C_{1} & R_{2}C_{2} \\ R_{3}C_{1} & R_{3}C_{2} \end{bmatrix} $$ **Example 3.** If $$ A = \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ and $$ B = \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 2\\ 1 & 2 & 0 \end{bmatrix} $$ , form the products AB and BA and show that AB ≠ BA. **Sol.** AB = $$ \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ $$ \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 2\\ 1 & 2 & 0 \end{bmatrix} $$ = $$ \begin{bmatrix} 1(1) -2(0) + 3 (1) & 1(0) -2(1) + 3(2) & 1(2) -2(2) + 3(0)\\ 2(1) + 3(0) -1(1) & 2(0) + 3(1) -1 (2) & 2(2) + 3(2) -1(0) \\ -3(1) +1(0) + 2(1) & -3(0) + 1(1) + 2(2) & -3(2) + 1 (2) + 2(0) \end{bmatrix} $$ = $$ \begin{bmatrix} 4 & 4 & -2\\ 1 & 1 & 10\\ -1 & 5 & -4 \end{bmatrix} $$ BA = $$ \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 2\\ 1 & 2 & 0 \end{bmatrix} $$ $$ \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ = $$ \begin{bmatrix} 1(1) + 0(2) + 2(-3) & 1(-2) + 0(3) + 2(1) & 1(3) + 0(-1) + 2(2)\\ 0(1) +1(2) + 2(-3) & 0(-2) + 1(3) + 2(1) & 0 (3) + 1(-1) + 2(2)\\ 1 (1) + 2 (2) + 0(-3) & 1(-2) + 2(3) + 0(1) & 1(3) +2(-1) + 0(2) \end{bmatrix} $$ = $$ \begin{bmatrix} -5 & 0 & 7\\ -4 & 5 & 3\\ 5 & 4 & 1 \end{bmatrix} $$ Orders of AB and BA are the same but their corresponding elements are not equal. Hence AB ≠ BA. **Example 4.** Show by means of an example that the matrix AB = O does not necessarily mean that either A = O or B = 0, where O stands for the null matrix. Sol. Consider the matrices $$ A = \begin{bmatrix} 1 & 1\\ 1 & 1\\ \end{bmatrix} $$ and $$ B = \begin{bmatrix} -1 & 0\\ 1 & 0\\ \end{bmatrix} $$ Here A ≠ 0, B ≠ 0. But AB = $$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} $$ $$ \begin{bmatrix} -1 & 0\\ 1 & 0\\ \end{bmatrix} $$ = $$ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=O $$ **Example 5.** Illustrate with the help of examples that multiplication of matrices is not commutative in general, i.e., AB ≠ BA. Point out the various possibilities. Sol. **Case I. AB is defined, but BA is not defined.** Let A be a matrix of order 2 x 3, and B be a matrix of order 3 x 4. Then AB is defined and is a matrix of order 2 x 4, whereas BA is not defined. **Case II. AB and BA are both defined but their orders are different.** Let A be a matrix of order 2 x 3 and B be a matrix of order 3 x 2. The product AB is defined and is a matrix of order 2 x 2. The product BA is defined and is a matrix of order 3 x 3. Since orders of AB and BA are different, AB ≠ BA.. **Case III. AB and BA are both defined and are matrices of the same order, yet AB ≠ BA (since corresponding elements are not equal).** let $$ A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} $$ and $$ \begin{bmatrix} 2 & 1\\ 4 & 1 \end{bmatrix} $$ Here AB and BA are both defined and are of order 2 x 2. AB = $$ \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} $$ $$ \begin{bmatrix} 2 & 1\\ 4 & 1 \end{bmatrix} $$= $$ \begin{bmatrix} 10 & 3\\ 22 & 7 \end{bmatrix} $$ BA = $$ \begin{bmatrix} 2 & 1\\ 4 & 1 \end{bmatrix} $$ $$ \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} $$= $$ \begin{bmatrix} 5 & 8\\ 7 & 12 \end{bmatrix} $$ Clearly, AB ≠ BA. Hence multiplication of matrices is not commutative in general, i.e., in general AB ≠ BA. Note. If AB = BA, we say 'A and B commute'. ### 3.6 Properties of Matrix Multiplication (i) Matrix multiplication is not commutative in general i.e., AB ≠ BA. (ii) Matrix multiplication is associative i.e., (AB)C = A(BC). (iii) Matrix multiplication is distributive with respect to matrix addition. i.e., A(B+C) = AB + AC. (iv) if A and I are square matrices of the same order, then AI = IA = A. (v) If A is a square matrix of order n, then A × A = A², Ax Ax A = A³, Ax Ax A ..... m times = A^m are all square matrices of order n. Also, we define A⁰ = I. (vi) For any positive integer n, Iⁿ = I. Note. If A is a square matrix such that A² = I, then A is called an involutory matrix. **Example 6.** Evaluate A²-3A+ 9I, if I is the unit matrix of order 3 and $$ A = \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ **Sol.** A² = A x A = $$ \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ $$ \begin{bmatrix} 1 & -2 & 3\\ 2 & 3 & -1\\ -3 & 1 & 2 \end{bmatrix} $$ = $$ \begin{bmatrix} 1(1)-2(2)+3(-3) & 1(-2)-2(3)+3(1) & 1(3)-2(-1)+3(2)\\ 2(1)+3(2)-1(-3) & 2(-2)+3(3)-1(1) & 2(3)+3(-1)-1(2)\\ -3(1)+1(2)+2(-3) & -3(-2)+1(3)+2(1) & -3(3)+1(-1)+2(2) \end{bmatrix}