11th Maths Weekly Test- 4 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test- 4 (Matrices and Determinants)-Aug 2020

Time : 1.30 Hours

Part - A

Generic 3 - Answer All Question

Identify the singular and non-singular matrices:\(\begin{bmatrix} 2&-3 &5 \\ 6 & 0 &4 \\ 1 & 5 & -7 \end{bmatrix}\)

Show that the points (a, b+c)(b, c+a) and (c, a+b) are collinear.

Show that \(\left| \begin{matrix} 1 & a & { a } \\ a & 1 & a \\ a & a & 1 \end{matrix} \right| ^{ 2 }=\left| \begin{matrix} 1-2{ a }^{ 2 } & -{ a }^{ 2 } & -{ a }^{ 2 } \\ -{ a }^{ 2 } & -1 & { a }^{ 2 }-2a \\ -{ a }^{ 2 } & { a }^{ 2 }-2a & -1 \end{matrix} \right| \)

If \(\begin{vmatrix} a & b &a\alpha +b \\ b & c & b\alpha+c\\ a\alpha+b & b \alpha+c &0 \end{vmatrix}=0.\)
prove that a, b, c are in G.P. or \(\alpha\) is a root of ax² + 2bx + c = 0.

If A=\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A³-6A²+7A+KI=O,find the value of k.

Determine the values of b so that the following matrices are singular:\(\begin{bmatrix}b-1 &2 &3 \\3 & 1 & 2 \\ 1 & -2 &4 \end{bmatrix}\)

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A+B)^T=A^T+B^T=B^T+A^T

Find the matrix A which satisfies the matrix relation A\(\begin{bmatrix} 1 & 2&3 \\4 & 5&6 \end{bmatrix}\)=\(\begin{bmatrix} -7 & -8&-9 \\2 & 4&6 \end{bmatrix}\)

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

If AB=A and BA=B, then show that A²=A and B²=B.

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11th Maths Weekly Test- 4 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test- 4 (Matrices and Determinants)-Aug 2020

Part - A

Generic 3 - Answer All Question

Identify the singular and non-singular matrices:\(\begin{bmatrix} 2&-3 &5 \\ 6 & 0 &4 \\ 1 & 5 & -7 \end{bmatrix}\)

Show that the points (a, b+c)(b, c+a) and (c, a+b) are collinear.

Show that \(\left| \begin{matrix} 1 & a & { a } \\ a & 1 & a \\ a & a & 1 \end{matrix} \right| ^{ 2 }=\left| \begin{matrix} 1-2{ a }^{ 2 } & -{ a }^{ 2 } & -{ a }^{ 2 } \\ -{ a }^{ 2 } & -1 & { a }^{ 2 }-2a \\ -{ a }^{ 2 } & { a }^{ 2 }-2a & -1 \end{matrix} \right| \)

If \(\begin{vmatrix} a & b &a\alpha +b \\ b & c & b\alpha+c\\ a\alpha+b & b \alpha+c &0 \end{vmatrix}=0.\) prove that a, b, c are in G.P. or \(\alpha\) is a root of ax2 + 2bx + c = 0.

If A=\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A3-6A2+7A+KI=O,find the value of k.

Determine the values of b so that the following matrices are singular:\(\begin{bmatrix}b-1 &2 &3 \\3 & 1 & 2 \\ 1 & -2 &4 \end{bmatrix}\)

If AT=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A+B)T=AT+BT=BT+AT

Find the matrix A which satisfies the matrix relation A\(\begin{bmatrix} 1 & 2&3 \\4 & 5&6 \end{bmatrix}\)=\(\begin{bmatrix} -7 & -8&-9 \\2 & 4&6 \end{bmatrix}\)

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

If AB=A and BA=B, then show that A2=A and B2=B.

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If \(\begin{vmatrix} a & b &a\alpha +b \\ b & c & b\alpha+c\\ a\alpha+b & b \alpha+c &0 \end{vmatrix}=0.\)
prove that a, b, c are in G.P. or \(\alpha\) is a root of ax² + 2bx + c = 0.

If A=\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A³-6A²+7A+KI=O,find the value of k.

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B= \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\),verify (A+B)^T=A^T+B^T=B^T+A^T

If AB=A and BA=B, then show that A²=A and B²=B.