Linear algebra: please solve q14 and 16 correctly and handwritten condition 5 are attached In Exercises 13-18, use condition (5) to determinewhether the given matrix Q is orthogonal.010-[!][22]13. Q =15. Q=14. Q =16. Q[√√3 1 √2]=1[27]11√5[23] DEFINITION 9Orthogonal MatricesA remarkable and useful fact about symmetric matrices is that they are always diago-nalizable. Moreover, the diagonalization of a symmetric matrix A can be accomplishedwith a special type of matrix known as an orthogonal matrix.EXAMPLE 5SolutionA real (n x n) matrix Q is called an orthogonal matrix if Q is invertible andQ-¹ = QT.4.7 Similarity Transformations and DiagonalizationDefinition 9 can be rephrased as follows: A real square matrix Q is orthogonal ifand only ifQ¹Q = 1.(4)Another useful description of orthogonal matrices can be obtained from Eq. (4). Inparticular, suppose that Q = [91, 92, ..., qn] is an (n × n) matrix. Since the ith row ofQ¹ is equal to q, the definition of matrix multiplication tells us:The ijth entry of QTQ is equal to qq.Therefore, by Eq. (4), an (n × n) matrix Q = [9₁, 92, ..., qn] is orthogonal if and onlyif:Q₁ =Verify that the matrices, Q₁ and Q2 are orthogonal:10 10 √20-1 0 11√√2The columns of Q, {91, 92,. .., qn},form an orthonormal set of vectors.01and Q₂ =010 01010=We use Eq. (4) to show that Q₁ is orthogonal. Specifically,1 0 -11 0 1of Q₁Q₁ =√20 √200-1 0 1We use condition Eq. (5) to show that Q₂ is orthogonal. The column vectors of Q₂ are,in the order they appear, {e2, e3, e₁}. Since these vectors are orthonormal, it followsfrom Eq. (5) that Q2 is orthogonal.1001233120 00200 0 2= I.(5)From the characterization of orthogonal matrices given in condition Eq. (5), thefollowing observation can be made: If Q = [q1, 92, … , ¶] is an (n × n) orthogonalmatrix and if P = [P₁, P2, ..., Pn] is formed by rearranging the columns of Q, then Pis also an orthogonal matrix.As a special case of this observation, suppose that P is a matrix formed by rear-ranging the columns of the identity matrix, I. Then, since I is an orthogonal matrix,it follows that P is orthogonal as well. Such a matrix P, formed by rearranging thecolumns of I, is called a permutation matrix. The matrix Q₂ in Example 5 is a specificinstance of a (3 x 3) permutation matrix.Orthogonal matrices have some special properties that make them valuable toolsfor applications. These properties were mentioned in Section 3.7 with regard to (2 x 2)orthogonal matrices. Suppose we think of an (n xn) matrix Q as defining a function (orlinear transformation) from R" to R". That is, for x in R", consider the function definedbyy = Qx.As the next theorem shows, if Q is orthogonal, then the function y = Qx preserves thelengths of vectors and the angles between pairs of vectors.

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Linear algebra: please solve q14 and 16 correctly and handwritten condition 5 are attached In Exercises 13-18, use condition (5) to determinewhether the given matrix Q is orthogonal.010-[!][22]13. Q =15. Q=14. Q =16. Q[√√3 1 √2]=1[27]11√5[23] DEFINITION 9Orthogonal MatricesA remarkable and useful fact about symmetric matrices is that they are always diago-nalizable. Moreover, the diagonalization of a symmetric matrix A can be accomplishedwith a special type of matrix known as an orthogonal matrix.EXAMPLE 5SolutionA real (n x n) matrix Q is called an orthogonal matrix if Q is invertible andQ-¹ = QT.4.7 Similarity Transformations and DiagonalizationDefinition 9 can be rephrased as follows: A real square matrix Q is orthogonal ifand only ifQ¹Q = 1.(4)Another useful description of orthogonal matrices can be obtained from Eq. (4). Inparticular, suppose that Q = [91, 92, ..., qn] is an (n × n) matrix. Since the ith row ofQ¹ is equal to q, the definition of matrix multiplication tells us:The ijth entry of QTQ is equal to qq.Therefore, by Eq. (4), an (n × n) matrix Q = [9₁, 92, ..., qn] is orthogonal if and onlyif:Q₁ =Verify that the matrices, Q₁ and Q2 are orthogonal:10 10 √20-1 0 11√√2The columns of Q, {91, 92,. .., qn},form an orthonormal set of vectors.01and Q₂ =010 01010=We use Eq. (4) to show that Q₁ is orthogonal. Specifically,1 0 -11 0 1of Q₁Q₁ =√20 √200-1 0 1We use condition Eq. (5) to show that Q₂ is orthogonal. The column vectors of Q₂ are,in the order they appear, {e2, e3, e₁}. Since these vectors are orthonormal, it followsfrom Eq. (5) that Q2 is orthogonal.1001233120 00200 0 2= I.(5)From the characterization of orthogonal matrices given in condition Eq. (5), thefollowing observation can be made: If Q = [q1, 92, … , ¶] is an (n × n) orthogonalmatrix and if P = [P₁, P2, ..., Pn] is formed by rearranging the columns of Q, then Pis also an orthogonal matrix.As a special case of this observation, suppose that P is a matrix formed by rear-ranging the columns of the identity matrix, I. Then, since I is an orthogonal matrix,it follows that P is orthogonal as well. Such a matrix P, formed by rearranging thecolumns of I, is called a permutation matrix. The matrix Q₂ in Example 5 is a specificinstance of a (3 x 3) permutation matrix.Orthogonal matrices have some special properties that make them valuable toolsfor applications. These properties were mentioned in Section 3.7 with regard to (2 x 2)orthogonal matrices. Suppose we think of an (n xn) matrix Q as defining a function (orlinear transformation) from R&#34; to R&#34;. That is, for x in R&#34;, consider the function definedbyy = Qx.As the next theorem shows, if Q is orthogonal, then the function y = Qx preserves thelengths of vectors and the angles between pairs of vectors.

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14. Q = /5 16. Q = 2= [₁ 1-2 21 32 -2 3