?It can be proved that a nonzero matrix ???? has rank k if and only if some \(k \times

In Exercises 1–2, find the rank and nullity of the matrix \(A\) by reducing it to row echelon form. (a) \(A=\left[\begin{array}{rrrr}1 & 2 & -1 & 1 \\2 & 4 & -2 & 2 \\3 & 6 & -3 & 3 \\4 & 8 & -4 & 4\end{array}\right]\) (b) \(A=\left[\begin{array}{rrrrr}1 & -2 & 2 & 3 & -1 \\-3 & 6 & -1 & 1 & -7 \\2 & -4 & 5 & 8 & -4\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A A=[_4 8 -4 4 ^3 6 -3 3 ^2 4 -2 2 ^1 2 -1 1] A=[_ 2 -4 5 8 -4 ^-3 6 -1 1 -7 ^1 -2 2 3 -1]

In Exercises 1–2, find the rank and nullity of the matrix \(A\) by reducing it to row echelon form. (a) \(A=\left[\begin{array}{rrrrr}1 & 0 & -2 & 1 & 0 \\0 & -1 & -3 & 1 & 3 \\-2 & -1 & 1 & -1 & 3 \\0 & 1 & 3 & 0 & -4\end{array}\right]\) (b) \(A=\left[\begin{array}{rrrr}1 & 3 & 1 & 3 \\0 & 1 & 1 & 0 \\-3 & 0 & 6 & -1 \\3 & 4 & -2 & 1 \\2 & 0 & -4 & -2\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A A=[ _0 1 3 0 -4 ^-2 -1 1-1 3 ^0 -1-3 1 3 ^1 0 -2 1 0] A=[ _ 2 0 -4 -2 ^3 4 -2 1 ^3 0 6 -1 ^0 1 1 0 ^1 3 1 3]

In Exercises 3–6, the matrix \(R\) is the reduced row echelon form of the matrix \(A\). a. By inspection of the matrix \(R\), find the rank and nullity of \(A\). b. Confirm that the rank and nullity satisfy Formula (4). c. Find the number of leading variables and the number of parameters in the general solution of \(A \mathbf{x}=0\) without solving the system. \(A=\left[\begin{array}{rrr}2 & -1 & -3 \\-1 & 2 & -3 \\1 & 1 & 4\end{array}\right] ; \quadR=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\) Equation Transcription: ???? []; [] Text Transcription: R A R A Ax=0 A=[ _1 1 4 ^-1 2 -3 ^2-1 -3]; R=[ _0 0 1 ^0 1 0 ^1 0 0]

In Exercises 3–6, the matrix \(R\) is the reduced row echelon form of the matrix \(A\). a. By inspection of the matrix \(R\), find the rank and nullity of \(A\). b. Confirm that the rank and nullity satisfy Formula (4). c. Find the number of leading variables and the number of parameters in the general solution of \(A \mathbf{x}=0\) without solving the system. \(A=\left[\begin{array}{rrr}2 & -1 & -3 \\-1 & 2 & -3 \\1 & 1 & -6\end{array}\right] ; \quadR=\left[\begin{array}{rrr}1 & 0 & -3 \\0 & 1 & -3 \\0 & 0 & 0\end{array}\right]\) Equation Transcription: ???? []; [] Text Transcription: R A R A Ax=0 A=[ _1 1 -6^-1 2 -3 ^2-1 -3]; R=[ _0 0 0 ^0 1 -3 ^1 0 -3]

In Exercises 3–6, the matrix \(R\) is the reduced row echelon form of the matrix \(A\). a. By inspection of the matrix \(R\), find the rank and nullity of \(A\). b. Confirm that the rank and nullity satisfy Formula (4). c. Find the number of leading variables and the number of parameters in the general solution of \(A \mathbf{x}=0\) without solving the system. \(A=\left[\begin{array}{rrr}2 & -1 & -3 \\-2 & 1 & 3 \\-4 & 2 & 6\end{array}\right] ; \quad R=\left[\begin{array}{rrr}1 & -\frac{1}{2} & -\frac{3}{2} \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right]\) Equation Transcription: ???? []; [] Text Transcription: R A R A Ax=0 A=[ _-4 2 -6^-2 1 -3 ^2-1 -3]; R=[ _0 0 0 ^0 1 -3 ^1 -1/2 -3/2]

In Exercises 3–6, the matrix \(R\) is the reduced row echelon form of the matrix \(A\). a. By inspection of the matrix \(R\), find the rank and nullity of \(A\). b. Confirm that the rank and nullity satisfy Formula (4). c. Find the number of leading variables and the number of parameters in the general solution of \(A \mathbf{x}=0\) without solving the system. \(A=\left[\begin{array}{rrrr}0 & 2 & 2 & 4 \\1 & 0 & -1 & -3 \\2 & 3 & 1 & 1 \\-2 & 1 & 3 &-2\end{array}\right] ; \quad R=\left[\begin{array}{rrrr}1 & 0 & -1 & 0 \\0 & 1 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0\end{array}\right]\) Equation Transcription: ???? [];[] Text Transcription: R A R A Ax=0 A=[ _-2 1 3 -2 ^2 3 1 1 ^1 0 -1 -3 ^ 0 2 2 4];R=[_0 0 0 0 ^0 0 0 0 ^0 1 1 0 ^1 0 -1 0]

In each part, find the largest possible value for the rank of \(A\) and the smallest possible value for the nullity of \(A\). a. \(A\) is \(4 \times 4\) b. \(A\) is \(3 \times 5\) c. \(A\) is \(5 \times 3\) Equation Transcription: is is is Text Transcription: A A A is 4 x 4 A is 3 x 5 A is 5 x 3

If \(A\) is an \(m \times n\) matrix, what is the largest possible value for its rank and the smallest possible value for its nullity? Equation Transcription: Text Transcription: A m x n

In each part, use the information in the table to: i. find the dimensions of the row space of \(A\), column space of \(A\), null space of \(A\), and null space of \(A^{T}\); ii. determine whether the linear system \(A \mathbf{x}=\mathbf{b}\) is consistent; iii. find the number of parameters in the general solution of each system in (ii) that is consistent. Equation Transcription: ????=???? Text Transcription: A A A A^T Ax=b

Verify that rank \((A)=\) rank \(\left(A^{T}\right)\). \(A=\left[\begin{array}{rrrr}1 & 2 & 4 & 0 \\-3 & 1 & 5 & 2 \\-2 & 3 & 9 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: (A) (A^T) A=[_-2 3 9 2 ^-3 1 5 2 ^1 2 4 0]

In Exercises 11–14 find the dimensions and bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{rr}1 & 4 \\0 & 3 \\-9 & 0\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_-9 0 ^0 3 ^1 4]

In Exercises 11–14 find the dimensions and bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{lll}1 & 2 & 4 \\2 & 4 & 8\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_2 4 8 ^1 2 4]

In Exercises 11–14 find the dimensions and bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{rrr}0 & -1 & -4 \\-1 & 0 & -4 \\-2 & 3 & 4\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_-2 3 4 ^-1 0 -4 ^0 -1 -4]

In Exercises 11–14 find the dimensions and bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{rrrr}3 & 4 & 0 & 7 \\1 & -5 & 2 & -2 \\-1 & 4 & 0 & -3 \\1 & -1 & 2 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_1 -1 2 2 ^-1 4 0 -3 ^3 4 0 7]

In Exercises 15–18 confirm the orthogonality statements in the two parts of Theorem 4.9.7 for the given matrix. The matrix in Exercise 11. \(A=\left[\begin{array}{rr}1 & 4 \\0 & 3 \\-9 & 0\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_-9 0 ^0 3 ^1 4]

In Exercises 15–18 confirm the orthogonality statements in the two parts of Theorem 4.9.7 for the given matrix. The matrix in Exercise 12. \(A=\left[\begin{array}{lll}1 & 2 & 4 \\2 & 4 & 8\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_2 4 8 ^1 2 4]

In Exercises 15–18 confirm the orthogonality statements in the two parts of Theorem 4.9.7 for the given matrix. The matrix in Exercise 13. \(A=\left[\begin{array}{rrr}0 & -1 & -4 \\-1 & 0 & -4 \\-2 & 3 & 4\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_-2 3 4 ^-1 0 -4 ^0 -1 -4]

In Exercises 15–18 confirm the orthogonality statements in the two parts of Theorem 4.9.7 for the given matrix. The matrix in Exercise 14. \(A=\left[\begin{array}{rrrr}3 & 4 & 0 & 7 \\1 & -5 & 2 & -2 \\-1 & 4 & 0 & -3 \\1 & -1 & 2 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_1 -1 2 2 ^-1 4 0 -3 ^3 4 0 7]

In Exercises 19–20 use the method of Example 5 to find bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{rrrr}0 & 2 & 8 & -7 \\2 & -2 & 4 & 0 \\-3 & 4 & -2 & 5\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[-3 4 -2 5 ^2 -2 4 0 60 2 8 -7]

In Exercises 19–20 use the method of Example 5 to find bases for the four fundamental spaces of the matrix. \(A=\left[\begin{array}{rrrrr}1 & 2 & 3 & 1 & 1 \\2 & 8 & 0 & 1 & 2 \\0 & 4 & -6 & 0 & 1 \\1 & 0 & 0 & 0 & 0\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_1 0 0 0 0 ^0 4 -6 0 1 ^2 8 0 1 2 ^1 2 3 1 1]

a. Find an equation relating nullity \((A)\) and nullity \(\left(A^{T}\right)\) for the matrix in Exercise 10. b. Find an equation relating nullity \((A)\) and nullity \((AA)\) for a general \(m \times n\) matrix. Equation Transcription: Text Transcription: (A) (A^T) (AA) m x n

Let \(T: R^{2} \rightarrow R^{3}\) be the linear transformation defined by the formula \(T\left(x_{1}, x_{2}\right)=\left(x_{1}+3 x_{2}, x_{1}-x_{2}, x_{1}\right)\) a. Find the rank of the standard matrix for \(T\). b. Find the nullity of the standard matrix for \(T\). Equation Transcription: Text Transcription: T:R^2 rightarrow R^3 T(x_1,x_2)=(x_1+3x_2,x1-x_2,x_1) T T

Let \(T: R^{5} \rightarrow R^{3}\) be the linear transformation defined by the formula \(T\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=\left(x_{1}+x_{2}, x_{2}+x_{3}+x_{4}, x_{4}+x_{5}\right)\) a. Find the rank of the standard matrix for \(T\). b. Find the nullity of the standard matrix for \(T\). Equation Transcription: Text Transcription: T:R^5 rightarrow R^3 T(x_1,x_2,x_3,x_4,x_5)=(x_1+x_2,x_2+x_3+x_4,x_4+x_5) T T

Discuss how the rank of \(A\) varies with \(t\). (a) \(A=\left[\begin{array}{lll}1 & 1 & t \\1 & t & 1 \\t & 1 & 1\end{array}\right]\) (b) \(A=\left[\begin{array}{rrr}t & 3 & -1 \\3 & 6 & -2 \\-1 & -3 & t\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A t A=[_t 1 1 ^1 t 1 ^1 1 t] A=[_-1-3 t ^3 6-2 ^t 3 -1]

Are there values of \(r\) and \(s\) for which \(\left[\begin{array}{ccc}1 & 0 & 0 \\0 & r-2 & 2 \\0 & s-1 & r+2 \\0 & 0 & 3\end{array}\right]\) has rank 1? Has rank 2? If so, find those values. Equation Transcription: [] Text Transcription: r s [ _0 0 3 ^0 s-1r+2 ^0 r-2 2 ^1 0 0]

a. Give an example of a \(3 \times 3\) matrix whose column space is a plane through the origin in 3-space. b. What kind of geometric object is the null space of your matrix? c. What kind of geometric object is the row space of your matrix? Equation Transcription: Text Transcription: 3 x 3

Suppose that \(A\) is a \(3 \times 3\) matrix whose null space is a line through the origin in 3-space. Can the row or column space of \(A\) also be a line through the origin? Explain. Equation Transcription: Text Transcription: A 3 x 3 A

a. If \(A\) is a \(3 \times 5\) matrix, then the rank of \(A\) is at most _______________. Why? b. If \(A\) is a \(3 \times 5\) matrix, then the nullity of \(A\) is at most _______________. Why? c. If \(A\) is a \(3 \times 5\) matrix, then the rank of \(A^{T}\) is at most _______________. Why? d. If \(A\) is a \(3 \times 5\) matrix, then the nullity of \(A^{T}\) is at most _______________. Why? Equation Transcription: Text Transcription: A 3x5 A A 3x5 A A 3x5 A^T A 3x5 A^T

a. If \(A\) is a \(3 \times 5\) matrix, then the number of leading 1’s in the reduced row echelon form of \(A\) is at most ______________. Why? b. If \(A\) is a \(3 \times 5\) matrix, then the number of parameters in the general solution of \(A \mathbf{x}=\mathbf{0}\) is at most ______________. Why? c. If \(A\) is a \(5 \times 3\) matrix, then the number of leading 1’s in the reduced row echelon form of \(A\) is at most ______________. Why? d. If \(A\) is a \(5 \times 3\) matrix, then the number of parameters in the general solution of \(A \mathbf{x}=\mathbf{0}\) is at most ______________. Why? Equation Transcription: ????=???? ????=???? Text Transcription: A 3x5 A A 3x5 Ax=0 A 5x3 A A 5x3 Ax=0

Let \(A\) be a \(7 \time 6\) matrix such that \(A \mathbf{x}=\mathbf{0}\) has only the trivial solution. Find the rank and nullity of \(A\). Equation Transcription: ????=???? Text Transcription: A 7x6 Ax=0 A

Let \(A\) be a \(5 \times 7\) matrix with rank 4. a. What is the dimension of the solution space of \(A \mathrm{x}=0\)? b. Is \(A \mathrm{x}=\mathbf{b}\) consistent for all vectors b in \(R^{5}\)? Explain. Equation Transcription: ????=???? ????=???? Text Transcription: A 5x7 Ax=0 Ax=b R^5

Let \(A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23}\end{array}\right]\) Show that \(A\) has rank 2 if and only if one or more of the following determinants is nonzero. \(\left|\begin{array}{ll}a_{11} & a_{12} \\a_{21} & a_{22}\end{array}\right|, \quad\left|\begin{array}{ll} a_{11} & a_{13} \\a_{21} & a_{23}\end{array}\right|, \quad\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{22} & a_{23}\end{array}\right|\) Equation Transcription: [] , , Text Transcription: A A=[_a_21 a_22 a_23 ^a_11 a_12 a_13] A=|a_21 a_22 ^a_11 a_12|, |_a_21 a_23 ^a_11 a_13|, | _a_22 a_23 ^a_12 a_13|

Use the result in Exercise 22 to show that the set of points \((x, y, z)\) in \(R^{3}\) for which the matrix \(\left[\begin{array}{lll}x & y & z \\1 & x & y\end{array}\right]\) has rank 1 is the curve with parametric equations \(x=t, \ y=t^{2}, \ z=t^{3}\). Equation Transcription: [] Text Transcription: (x,y,z) R^3 [_1 x y ^x y z] x=t, y=t^2, z=t^3

Find matrices \(A\) and \(B\) for which rank \((A)\) = rank \((B)\), but rank \(\left(A^{2}\right) \neq\) rank \(\left(B^{2}\right)\). Equation Transcription: Text Transcription: A B A B (A2)neq (B2)

In Example 6 of Section 4.7 we showed that the row space and the null space of the matrix \(A=\left[\begin{array}{rrrrrr}1 & 3 & -2 & 0 & 2 & 0 \\2 & 6 & -5 & -2 & 4 & -3 \\0 & 0 & 5 & 10 & 0 & 15 \\2 & 6 & 0 & 8 & 4 & 18\end{array}\right]\) are orthogonal complements in \(R^{6}\), as guaranteed by part (a) of Theorem 4.9.7. Show that null space of \(A^{T}\) and the column space of \(A\) are orthogonal complements in \(R^{4}\), as guaranteed by part (b) of Theorem 4.9.7. [Suggestion: Show that each column vector of \(A\) is orthogonal to each vector in a basis for the null space of \(A^{T}\). Equation Transcription: [] Text Transcription: A=[ _2 6 0 8 4 18 ^0 0 5 10 0 15 ^2 6 -5 -2 4 -3 ^1 3- 2 0 2 0] R^6 A^T A R^4 A A^T

Confirm the results stated in Theorem 4.9.7 for the matrix. \(A=\left[\begin{array}{rrrrr}-2 & -5 & 8 & 0 & -17 \\1 & 3 & -5 & 1 & 5 \\3 & 11 & -19 & 7 & 1 \\1 & 7 & -13 & 5 & -3\end{array}\right]\) Equation Transcription: [] Text Transcription: A=[_17 -13 5 -3 ^3 11 -19 7 1 ^1 3 -5 1 5 ^-2 -5 8 0-17]

In each part, state whether the system is overdetermined or underdetermined. If overdetermined, find all values of the \(b^{\prime}\) s for which it is inconsistent, and if underdetermined, find all values of the \(b^{\prime}\) s for which it is inconsistent and all values for which it has infinitely many solutions. (a) \(\left[\begin{array}{rr}1 & -1 \\-3 & 1 \\0 & 1\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}b_{1} \\b_{2} \\b_{3}\end{array}\right]\) (b) \(\left[\begin{array}{rrr}1 & -3 & 4 \\-2 & -6 & 8\end{array}\right]\left[\begin{array}{l}x \\y \\z\end{array}\right]=\left[\begin{array}{l}b_{1} \\b_{2}\end{array}\right]\) (c) \(\left[\begin{array}{rrr}1 & -3 & 0 \\-1 & 1 & 1\end{array}\right]\left[\begin{array}{l}x \\y \\z\end{array}\right]=\left[\begin{array}{l}b_{1} \\b_{2}\end{array}\right]\) Equation Transcription: [][]=[] [][]=[] [][]=[] Text Transcription: b's b's [ _0 1 ^-3 1 ^1 -1][_y ^x]=[ _b_3 ^b_2 ^b_1] [_-2 -6 8 ^1-3 4][ _z ^y ^x]=[_b_2 ^b_1] [_-1 1 1 ^1-3 0][ _z ^y ^x]=[_b_2 ^b_1]

What conditions must be satisfied by \(b_{1}, b_{2}, b_{3}, b_{4}\), and \(b_{5}\) for the overdetermined linear system \(x_{1}-3 x_{2}=b_{1}\) \(x_{1}+2 x_{2}=b_{2}\) \(x_{1}+x_{2}=b_{3}\) \(x_{1}-4 x_{2}=b_{4}\) \(x_{1}+5 x_{2}=b_{5}\) to be consistent? Equation Transcription: Text Transcription: b_1,b_2,b_3,b_4 b_5 x_1-3x_2=b_1 x_1+2x_2=b_2 x_1+x_2=b_3 x_1-4x_2=b_4 x_1+5x_2=b_5

Prove: If \(k \neq 0\) , then \(A\) and \(kA\) have the same rank. Equation Transcription: Text Transcription: k neq 0 A kA

Prove: If a matrix \(A\) is not square, then either the row vectors or the column vectors of \(A\) are linearly dependent. Equation Transcription: Text Transcription: A A

Use Theorem 4.9.3 to prove Theorem 4.9.4.

Prove Theorem 4.9.7(b).

Prove: If a vector \(v\) in \(R^{n}\) is orthogonal to each vector in a basis for a subspace \(W\) of \(R^{n}\), then v is orthogonal to every vector in \(W\). Equation Transcription: Text Transcription: v R^n W R^n v W

Prove: (q) implies (b) in Theorem 4.9.8.

In parts (a)–( j) determine whether the statement is true or false, and justify your answer. Either the row vectors or the column vectors of a square matrix are linearly independent.

In parts (a)–( j) determine whether the statement is true or false, and justify your answer. A matrix with linearly independent row vectors and linearly independent column vectors is square.

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. The nullity of a nonzero \(m \times n\) matrix is at most m. Equation Transcription: Text Transcription: (a)–( j) m x n

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. Adding one additional column to a matrix increases its rank by one. Equation Transcription: Text Transcription: (a)–( j)

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. The nullity of a square matrix with linearly dependent rows is at least one. Equation Transcription: Text Transcription: (a)–( j)

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. If ???? is square and \(????x = b\) is inconsistent for some vector b, then the nullity of ???? is zero. Equation Transcription: Text Transcription: (a)–( j) ????x = b

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. If a matrix ???? has more rows than columns, then the dimension of the row space is greater than the dimension of the column space. Equation Transcription: Text Transcription: (a)–( j)

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. If rank \(\left(A^{T}\right)=\operatorname{rank}(A)\), then ???? is square. Equation Transcription: (????????) = rank(????) Text Transcription: (a)–( j) (????^????) = rank(????)

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. There is no \(3 \times 3\) matrix whose row space and null space are both lines in 3-space. Equation Transcription: Text Transcription: (a)–( j) 3 × 3

In parts \((a)–( j)\) determine whether the statement is true or false, and justify your answer. If ???? is a subspace of \(R^{\mathrm{n}}\) and ???? is a subspace of ????, then \(W^{\perp}\) is a subspace of \(V^{\perp}\). Equation Transcription: ????n ????? ????? Text Transcription: (a)–( j) R^n W^perp V^perp

It can be proved that a nonzero matrix ???? has rank k if and only if some \(k \times k\) submatrix has a nonzero determinant and all square submatrices of larger size have determinant zero. Use this fact to find the rank of \(A=\left|\begin{array}{ccccc} 3 & -1 & 3 & 2 & 5 \\ 5 & -3 & 2 & 3 & 4 \\ 1 & -3 & -5 & 0 & -7 \\ 7 & -5 & 1 & 4 & 1 \end{array}\right| \) Check your result by computing the rank of ???? in a different way. Equation Transcription: Text Transcription: k x k A=|3 -1 3 2 5 5 -3 2 3 4 1 -3 -5 0 -7 7 -5 1 4 1|

Sylvester’s inequality states that if ???? and ???? are \(n \times n\) matrices with rank \(r_{A}\) and \(r_{B}\), respectively, then the rank \(r_{AB}\) of ???????? satisfies the inequality \(\mathrm{r}_{A}+\mathrm{r}_{B}-\mathrm{n} \leq \mathrm{r}_{A B} \leq \min \left(\mathrm{r}_{A}, \mathrm{r}_{B}\right)\) where min \(\left(r_{A}, r_{B}\right)\) denotes the smaller of \(r_{A}\) and \(r_{B}\) or their common value if the two ranks are the same. Use your technology utility to confirm this result for some matrices of your choice. Equation Transcription: r???? r???? r???????? r???? + r???? ? n r???????? min(r????, r????) (r????, r????) Text Transcription: n x n r_???? r_???? r_???????? r_???? + r_???? ? n leq r_???????? leq min(r_????, r_????) (r_????, r_????)