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Name Grade &Section Roll No 1.Given M=[} x-4&6 4&x-6 ] ,if vert Mvert =0 then the value o fxis A. 0 B. 10 -10 D. A&B 2. If matrix B is a 3times 3 non singular matrix and vert Bvert =-2 ,then vert -3Bvert is equal to? A. 54 B. -6 C. -54 D. 6 E -16 3.Given M=(} -1&-2&-3 0&-3&-4 1&1&0 ) ,which elementary row operation is most approprate to get (} -1&-2&-3 -1&1&1 1&1&0 ) from the given matrix M? A. R_(2)arrow R_(2)+R_(1) B. R_(2)arrow R_(2)-R_(1) C R_(2)arrow R_(1)-R_(2) D. R_(2)arrow R_(2)-R_(3) Which one of the ff matrix is in Reduced Row Echelon form? D (} 1&0&0 0&0&0 0&1&0 ) A. (} -1&0&1 0&0&0 0&1&0 ) B. (} 1&0&0 0&0&1 0&1&0 ) C. (} 1&0&12 0&1&0 0&0&0 ) A is an invertible 3times 3 matrix ,which one is false about matrix A? B. It is a singular 'matrix A. Its determinant is different from zero C. Its inverse is unique D vert A^2vert =vert Avert ast vert Avert E.All are true ich system of linear equation has No solution? C. ) x+2y+3z=5 3x+6y+9z=3 x+2y+8z=1 A. ) x+y+2z=1 y+2x+z=3 x+2y+2z=0 B. ) x-y+z=0 2y-z=3

Pergunta

Name	Grade &Section	Roll No
1.Given
M=[} x-4&6 4&x-6 ] ,if vert Mvert =0 then the value o fxis
A. 0
B. 10
-10
D. A&B
2. If matrix B is a 3times 3 non singular matrix and vert Bvert =-2 ,then vert -3Bvert  is equal to?
A. 54 B. -6
C. -54
D. 6
E -16
3.Given
M=(} -1&-2&-3 0&-3&-4 1&1&0 ) ,which elementary row operation is most approprate
to get
(} -1&-2&-3 -1&1&1 1&1&0 )
from the given matrix M?
A. R_(2)arrow R_(2)+R_(1)
B. R_(2)arrow R_(2)-R_(1)
C R_(2)arrow R_(1)-R_(2)
D. R_(2)arrow R_(2)-R_(3)
Which one of the ff matrix is in Reduced Row Echelon form?
D (} 1&0&0 0&0&0 0&1&0 )
A. (} -1&0&1 0&0&0 0&1&0 )
B. (} 1&0&0 0&0&1 0&1&0 )
C. (} 1&0&12 0&1&0 0&0&0 )
A is an invertible
3times 3
matrix ,which one is false about matrix A?
B. It is a singular 'matrix
A. Its determinant is different from zero
C. Its inverse is unique
D vert A^2vert =vert Avert ast vert Avert 
E.All are true
ich system of linear equation has No solution?
C.  ) x+2y+3z=5 3x+6y+9z=3 x+2y+8z=1 
A.
 ) x+y+2z=1 y+2x+z=3 x+2y+2z=0 
B.  ) x-y+z=0 2y-z=3

Name Grade &Section Roll No 1.Given M=[} x-4&6 4&x-6 ] ,if vert Mvert =0 then the value o fxis A. 0 B. 10 -10 D. A&B 2. If matrix B is a 3times 3 non singular matrix and vert Bvert =-2 ,then vert -3Bvert is equal to? A. 54 B. -6 C. -54 D. 6 E -16 3.Given M=(} -1&-2&-3 0&-3&-4 1&1&0 ) ,which elementary row operation is most approprate to get (} -1&-2&-3 -1&1&1 1&1&0 ) from the given matrix M? A. R_(2)arrow R_(2)+R_(1) B. R_(2)arrow R_(2)-R_(1) C R_(2)arrow R_(1)-R_(2) D. R_(2)arrow R_(2)-R_(3) Which one of the ff matrix is in Reduced Row Echelon form? D (} 1&0&0 0&0&0 0&1&0 ) A. (} -1&0&1 0&0&0 0&1&0 ) B. (} 1&0&0 0&0&1 0&1&0 ) C. (} 1&0&12 0&1&0 0&0&0 ) A is an invertible 3times 3 matrix ,which one is false about matrix A? B. It is a singular 'matrix A. Its determinant is different from zero C. Its inverse is unique D vert A^2vert =vert Avert ast vert Avert E.All are true ich system of linear equation has No solution? C. ) x+2y+3z=5 3x+6y+9z=3 x+2y+8z=1 A. ) x+y+2z=1 y+2x+z=3 x+2y+2z=0 B. ) x-y+z=0 2y-z=3

Solução

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FelipeProfissional · Tutor por 6 anos

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Let's solve each question step by step:<br /><br />1. Given \( M = \begin{pmatrix} x-4 & 6 \\ 4 & x-6 \end{pmatrix} \), if \( \vert M \vert = 0 \), then the value of \( x \) is:<br /> The determinant of matrix \( M \) is given by:<br /> \[<br /> \vert M \vert = (x-4)(x-6) - (6 \cdot 4)<br /> \]<br /> Simplifying this, we get:<br /> \[<br /> \vert M \vert = x^2 - 10x + 24 - 24 = x^2 - 10x<br /> \]<br /> Setting the determinant to zero:<br /> \[<br /> x^2 - 10x = 0<br /> \]<br /> Factoring out \( x \):<br /> \[<br /> x(x - 10) = 0<br /> \]<br /> Therefore, \( x = 0 \) or \( x = 10 \). So the correct answer is:<br /> D. A & B<br /><br />2. If matrix \( B \) is a \( 3 \times 3 \) non-singular matrix and \( \vert B \vert = -2 \), then \( \vert -3B \vert \) is equal to:<br /> For any scalar \( k \) and an \( n \times n \) matrix \( B \):<br /> \[<br /> \vert kB \vert = k^n \vert B \vert<br /> \]<br /> Here, \( k = -3 \) and \( n = 3 \):<br /> \[<br /> \vert -3B \vert = (-3)^3 \vert B \vert = -27 \cdot (-2) = 54<br /> \]<br /> So the correct answer is:<br /> A. 54<br /><br />3. Given \( M = \begin{pmatrix} -1 & -2 & -3 \\ 0 & -3 & -4 \\ 1 & 1 & 0 \end{pmatrix} \), which elementary row operation is most appropriate to get \( \begin{pmatrix} -1 & -2 & -3 \\ -1 & 1 & 1 \\ 1 & 1 & 0 \end{pmatrix} \)?<br /> To transform the second row from \( \begin{pmatrix} 0 & -3 & -4 \end{pmatrix} \) to \( \begin{pmatrix} -1 & 1 & 1 \end{pmatrix} \), we can add the first row to the second row:<br /> \[<br /> R_2 \rightarrow R_2 + R_1<br /> \]<br /> So the correct answer is:<br /> A. \( R_2 \rightarrow R_2 + R_1 \)<br /><br />4. Which one of the following matrices is in Reduced Row Echelon Form?<br /> Reduced Row Echelon Form (RREF) requires that:<br /> - Each leading entry in a row is 1.<br /> - Each leading 1 is the only nonzero entry in its column.<br /> - The leading 1 in each row is to the right of the leading 1 in the previous row.<br /> - Rows with all zero elements are at the bottom.<br /><br /> Let's check each option:<br /> - Option A: \( \begin{pmatrix} -1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \) is not in RREF because the leading entry in the first row is not 1.<br /> - Option B: \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \) is not in RREF because the leading 1 in the third row is not to the right of the leading 1 in the second row.<br /> - Option C: \( \begin{pmatrix} 1 & 0 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \) is in RREF.<br /><br /> So the correct answer is:<br /> C. \( \begin{pmatrix} 1 & 0 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \)<br /><br />5. If \( A \) is an invertible \( 3 \times 3 \) matrix, which one is false about matrix \( A \)?<br /> - A. Its determinant is different from zero (True, since an invertible matrix has a non-zero determinant).<br /> - B. It is a singular matrix (False, since a singular matrix is not invertible).<br /> - C. Its inverse is unique (True, every invertible matrix has a unique inverse).<br /> - D. \( \vert A^2 \vert = \vert A \vert \ast \vert A \vert \) (True, as the determinant of a product of matrices equals the product of their determinants).<br /><br /> So the correct answer is:<br /> B. It is a singular matrix<br /><br />6. Which system of linear equations has no solution?<br /> - System A: \( \begin{cases} x + y + 2z = 1 \\ y + 2x + z = 3 \\ x + 2y + 2z = 0 \end{cases} \)<br /> This system may have a solution; further analysis is needed.<br /> - System B: \( \begin{cases} x - y + z = 0 \\ 2y - z = 3 \end{cases} \)<br /> This system may have a solution; further analysis is needed.<br /> - System C: \( \begin{cases} x + 2y + 3z = 5 \\ 3x + 6y + 9z = 3 \\ x + 2y + 8z = 1 \end{cases} \)<br /> This system is inconsistent because the second equation is a multiple of the first but with a different constant term, indicating parallel planes that do not intersect.<br /><br /> So the correct answer is:<br /> C. \( \begin{cases} x + 2y + 3z = 5 \\ 3x + 6y + 9z = 3 \\ x + 2y + 8z = 1 \end{cases} \)
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