Let A = [[1, 0, 0], [2, 1, 0], [3, 2,...

MathematicsAdjoint and Inverse of a MatrixJEE Advanced 2006Moderate

Comprehension Passage

Let

A = 123012001

, and

U_{1}, U_{2}

and

U_{3}

are columns of a

3 \times 3

matrix

U

. If column matrices

U_{1}, U_{2}

and

U_{3}

satisfying

A U_{1} = 100, A U_{2} = 230, A U_{3} = 231

evaluate as directed in the following questions.

Question 1:

The value $∣ U ∣$ is

(A)

(B)

-3

(C)

3/2

(D)

Answer:A

Question 2:

The sum of the elements of the matrix $U^{- 1}$ is

(A)

-1

(B)

(C)

(D)

Answer:B

Question 3:

The value of $[320] U 320$ is

(A)

(B)

5/2

(C)

(D)

3/2

Answer:A

Visualized Solution (English)

Defining the Matrix Equation

Given matrix $A = 123012001$
Let $U = [U_{1} U_{2} U_{3}]$ be a $3 \times 3$ matrix.
We are given $A U_{1}, A U_{2},$ and $A U_{3}$ .
By the property of matrix multiplication: $A U = [A U_{1} A U_{2} A U_{3}]$

Constructing Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A U

Substitute the given column vectors into the matrix $A U$ :
$A U = 100230231$
Note: This is an upper triangular matrix.

Applying Determinant Properties

To find $∣ U ∣$ , use the property: $∣ A U ∣ = ∣ A ∣ \cdot ∣ U ∣$
Therefore, $∣ U ∣ = \frac{∣ A U ∣}{∣ A ∣}$
We need to calculate $∣ A ∣$ and $∣ A U ∣$ separately.

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣ A ∣ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣ A U ∣

Since $A$ is lower triangular: $∣ A ∣ = 1 \times 1 \times 1 = 1$
Since $A U$ is upper triangular: $∣ A U ∣ = 1 \times 3 \times 1 = 3$
Using $∣ A ∣ \cdot ∣ U ∣ = ∣ A U ∣ ⟹ 1 \cdot ∣ U ∣ = 3$
Final value: $∣ U ∣ = 3$

Finding Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U

To find $U$ , solve $U = A^{- 1} 100230231$
Calculating $A^{- 1}$ : $A^{- 1} = 1 - 2 1 01 - 2 001$
Perform matrix multiplication: $U = 1 - 2 1 2 - 1 - 4 2 - 1 - 3$

Sum of Elements of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U^{- 1}

Calculate $U^{- 1} = \frac{1}{∣ U ∣} adj (U)$
$U^{- 1} = \frac{1}{3} - 1 - 7 9 - 2 - 5 6 0 - 3 3$
Sum of elements $= \frac{1}{3} (- 1 - 2 + 0 - 7 - 5 - 3 + 9 + 6 + 3) = 0$

Final Quadratic Form Evaluation

Evaluate $X^{T} U X$ where $X = 320$
First, $U X = 1 - 2 1 2 - 1 - 4 2 - 1 - 3 320 = 7 - 8 - 5$
Then, $[320] 7 - 8 - 5 = 21 - 16 + 0 = 5$

Key Takeaways

Key Takeaway 1: Matrix $U$ can be built column-wise if $A U_{i}$ is known.
Key Takeaway 2: Determinant of triangular matrices is the product of diagonal elements.
Next Challenge: What happens if matrix $A$ is singular? Can we still find a unique $U$ ?

00:00 / 00:00

Conceptually Similar Problems

MathematicsAlgebraic Operations on MatricesJEE Main 2012Easy

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Let $A = 123012001$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = 100$ and $A u_{2} = 010$ , then $u_{1} + u_{2}$ is equal to:

(A)

- 1 10

(B)

- 1 1 - 1

(C)

- 1 - 1 0

(D)

1 - 1 - 1

Answer:D

MathematicsAlgebraic Operations on MatricesJEE Advanced 2011Easy

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Let M be a $3 \times 3$ matrix satisfying $M 010 = - 1 23, M 1 - 1 0 = 11 - 1$ , and $M 111 = 0012$ . Then the sum of the diagonal entries of M is

Answer:9

MathematicsProperties of DeterminantsJEE Advanced 2003Moderate

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If matrix $A = a b c b c a c a b$ where $a, b, c$ are real positive numbers, $ab c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .

Answer:4

MathematicsAlgebraic Operations on MatricesJEE Advanced 2016Moderate

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Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

(A)

(a) 52

(B)

(b) 103

(C)

(D)

(d) 205

Answer:B

MathematicsAdjoint and Inverse of a MatrixJEE Main 2004Easy

View in:English Hindi

Let $A = 121 - 1 11 1 - 3 1$ and $10 B = 4 - 5 1 20 - 2 2 α 3$ . If $B$ is the inverse of matrix $A$ , then $α$ is

(A)

5

(B)

- 1

(C)

2

(D)

- 2

Answer:A

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 2009Moderate

View in:English Hindi

Comprehension Passage

Let

A

be the set of all

3 \times 3

symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Question 1:

The number of matrices in $A$ is

(A)

(B)

(C)

(D)

Answer:A

Question 2:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

(A)

less than 4

(B)

at least 4 but less than 7

(C)

at least 7 but less than 10

(D)

at least 10

Answer:B

Question 3:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

(A)

(B)

more than 2

(C)

(D)

Answer:B

MathematicsAlgebraic Operations on MatricesJEE Advanced 2012Easy

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If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I$ , where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = x y z \neq = 000$ such that

(A)

(a)

P X = 000

(B)

(b)

P X = X

(C)

(c)

P X = 2 X

(D)

(d)

P X = - X

Answer:D

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Let $P = 323 - 1 0 - 5 - 2 α 0$ , where $α \in R$ . Suppose $Q = [q_{ij}]$ is a matrix such that $P Q = k I$ , where $k \in R, k \neq = 0$ and $I$ is the identity matrix of order $3$ . If $q_{23} = - k /8$ and $det (Q) = k^{2} /2$ , then

* Multiple Correct Options

(A)

α = 0, k = 8

(B)

4 α - k + 8 = 0

(C)

det (P adj (Q)) = 2^{9}

(D)

det (Q adj (P)) = 2^{13}

Answer:B, C

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Let $u = \hat{i} + \hat{j}, v = \hat{i} - \hat{j}$ and $w = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . If $\overset{n}{^}$ is a unit vector such that $u \cdot \overset{n}{^} = 0$ and $v \cdot \overset{n}{^} = 0$ , then $∣ w \cdot \overset{n}{^} ∣$ is equal to

(A)

(B)

(C)

(D)

Answer:A

MathematicsProperties of DeterminantsJEE Advanced 2012Easy

View in:English Hindi

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

(A)

(a)

2^{10}

(B)

(b)

2^{11}

(C)

(c)

2^{12}

(D)

(d)

2^{13}

Answer:D

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value ∣U∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the elements of the matrix U−1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of [3​2​0​]U​320​​ is

Visualized Solution (English)

Defining the Matrix Equation

Constructing Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AU

Applying Determinant Properties

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣A∣ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣AU∣

Finding Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U

Sum of Elements of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U−1

Final Quadratic Form Evaluation

Key Takeaways

Conceptually Similar Problems

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=​1416​014​001​​ and I be the identity matrix of order 3. If Q=[qij​] is a matrix such that P50−Q=I, then q21​q31​+q32​​ equals

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Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value ∣U∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the elements of the matrix U−1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of [3​2​0​]U​320​​ is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

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The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

Constructing Matrix $A U$

Calculating $∣ A ∣$ and $∣ A U ∣$

Finding Matrix $U$

Sum of Elements of $U^{- 1}$

Let $A = 123012001$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = 100$ and $A u_{2} = 010$ , then $u_{1} + u_{2}$ is equal to:

If matrix $A = a b c b c a c a b$ where $a, b, c$ are real positive numbers, $ab c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .

Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

Let $A = 121 - 1 11 1 - 3 1$ and $10 B = 4 - 5 1 20 - 2 2 α 3$ . If $B$ is the inverse of matrix $A$ , then $α$ is

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

Let $u = \hat{i} + \hat{j}, v = \hat{i} - \hat{j}$ and $w = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . If $\overset{n}{^}$ is a unit vector such that $u \cdot \overset{n}{^} = 0$ and $v \cdot \overset{n}{^} = 0$ , then $∣ w \cdot \overset{n}{^} ∣$ is equal to

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

The value $∣ U ∣$ is

The sum of the elements of the matrix $U^{- 1}$ is

The value of $[320] U 320$ is

Let $A = 123012001$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = 100$ and $A u_{2} = 010$ , then $u_{1} + u_{2}$ is equal to:

If matrix $A = a b c b c a c a b$ where $a, b, c$ are real positive numbers, $ab c = 1$ and $A^{T} A = I$ , then find the value of $a^{3} + b^{3} + c^{3}$ .

Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

Let $A = 121 - 1 11 1 - 3 1$ and $10 B = 4 - 5 1 20 - 2 2 α 3$ . If $B$ is the inverse of matrix $A$ , then $α$ is

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

Let $u = \hat{i} + \hat{j}, v = \hat{i} - \hat{j}$ and $w = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . If $\overset{n}{^}$ is a unit vector such that $u \cdot \overset{n}{^} = 0$ and $v \cdot \overset{n}{^} = 0$ , then $∣ w \cdot \overset{n}{^} ∣$ is equal to

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is