Matrices V Lowersixth Science Further Mathematics

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Challenge on inverses, Gauss, and Cramer's rule.

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Matrices V Lowersixth Science Further Mathematics


Froggy Jumps Matrices V Lowersixth Science Further MathematicsOnline version Challenge on inverses, Gauss, and Cramer's rule. by YAKILI LMS 1 What is the inverse of a 2x2 matrix [a b; c d] if ad - bc ≠ 0? a A^{-1} = (1/(ad-bc)) [ d -b; -c a ] b A^{-1} = (1/(ad+bc)) [ d b; c a ] c A^{-1} = [ a b; c d ] 2 Which property holds for any invertible matrix A? a A^2 = I b A A^{-1} = I and A^{-1} A = I c AA^T = I 3 If A B = I, what is B? a B = I - A b B = A^{-1} c B = A^T 4 The determinant of A is zero. What can be said about A^{-1}? a A^{-1} exists but not unique b A^{-1} does not exist c A^{-1} = 0 matrix 5 Which matrix operation gives the inverse of a 2x2 matrix [a b; c d]? a Swap a and b, divide by a+d b Swap a and c, divide by ac-bd c Swap a and d, negate b and c, divide by ad-bc 6 Cramer's rule applies to systems with how many solutions in terms of determinant? a No solution when det(A) ≠ 0 b Infinite solutions when det(A) ≠ 0 c Unique solution when det(A) ≠ 0 7 Gaussian elimination transforms a system into which form? a Row-echelon form or reduced row-echelon form b Diagonal form only c Identity matrix form only 8 What does Gauss-Jordan elimination produce for the augmented matrix? a Reduced row-echelon form b Triangular form c Diagonal matrix only 9 To solve Ax = b by Gaussian elimination, you augment A with b. a False b Only if det(A) ≠ 0 c True 10 Cramer's rule requires which condition on A for a unique solution? a det(A) ≠ 0 b A is symmetric c A is diagonal 11 If x_i is a solution via Cramer's rule, what is used in the numerator? a Determinant of A_i obtained by replacing i-th column with b b Determinant of A c Determinant of A inverse 12 What is the property of (A^{-1})^{-1}? a A b I c A^T 13 The product of A and its inverse equals: a Diagonal matrix without ones b Zero matrix c Identity matrix 14 If A is 3x3 and det(A) = 2, what is det(A^{-1})? a -2 b 1/2 c 2 15 Which method finds x in Ax=b without matrix inversion? a Cramer's rule b Matrix transposition c Gaussian elimination 16 Which statement is true about inverse of A? a A^{-1} = A b A A^T = I c A A^{-1} = I 17 Which of the following is a necessary condition for A to be invertible? a A has a zero row only b det(A) ≠ 0 c rank(A) = 0 18 Cramer’s rule solves for x1 in which form? a Det(A1) + Det(A) b Det(A1)/Det(A) c Det(A)/Det(A1) 19 What is the effect of row operations on det(A)? a Det becomes zero after any operation b Det changes by a factor if rows are scaled c Det stays constant always 20 To check invertibility, you check if det(A) is: a Nonzero b Zero only c Positive 21 Which is not a standard row operation? a Mixing rows without updating determinants b Multiply a row by a nonzero scalar c Swap two rows 22 In Ax=b, if A is 2x2, how many pivots maximum? a 1 b 2 c 0 23 What is the identity matrix of size n? a All ones b Diagonal with 1s on main diagonal c All zeros 24 What does det(A) measure in a linear system? a Number of solutions b Volume scaling factor of the linear transformation c Matrix symmetry 25 If A is invertible, what can be said about Ax = b? a Multiple solutions for all b b No solution for some b c A unique solution exists for all b 26 What is a characteristic of a singular matrix? a Determinant is negative only b Determinant is zero c Determinant is nonzero 27 Which matrix operation yields A^{-1} if possible? a Replace with A^2 b Compute adjugate and divide by det(A) c Take transpose only 28 In 2x2, if det(A) = -3, det(A^{-1}) is: a 3 b -3 c -1/3 29 Gauss elimination aims to produce: a A symmetric form b An upper triangular form c A diagonal form only 30 What is required to apply Cramer's rule for a 3x3 system? a Det(A) ≠ 0 b A is diagonal c A is symmetric 31 The inverse of a matrix A is: a A^{-1} A = 0 b The matrix that satisfies A A^{-1} = I c The transpose of A 32 What is A^{-1} used to compute in Ax=b? a x = A b b x = A^{-1} b c x cannot be computed with A^{-1} 33 Which method is often used to solve Ax=b without calculating A^{-1}? a LU decomposition or Gaussian elimination b Matrix squaring c Eigenvalue decomposition 34 What does det(A) ≠ 0 imply about the linear transformation? a Transformation is symmetric b Transformation is invertible c Transformation collapses dimension to zero 35 Cramer's rule expresses x_i as a ratio of: a Det(A_i) / Det(A) b Det(A) / Det(A_i) c Sum of determinants 36 Which is not a typical property of A^{-1}? a A^{-1} A = I b (A^{-1})^{-1} = A c A^{-1} is always symmetric 37 If A is invertible, then the system Ax=b has: a Infinite solutions for all b b No solution for some b c A unique solution for every b 38 What form does Gauss elimination aim to achieve for the coefficient matrix? a Diagonal with equal entries b Upper triangular c Lower triangular 39 What does the augmented matrix [A\|b] represent? a A times b b A plus b c The system Ax=b 40 In a 3x3 system with det(A) ≠ 0, how many pivots exist? a 1 b 3 c 2 41 What is the effect of swapping two rows on det(A)? a Det(A) changes sign b Det(A) doubles c Det(A) remains same 42 What is the central idea of Cramer's rule? a Solve for each variable via determinants b Use row reduction only c Compute inverse exclusively 43 If A is 2x2 with det(A)=4, det(A^{-1}) equals: a 4 b -4 c 1/4 44 Which operation preserves the solution set of Ax=b? a Row operations on augmented matrix b Multiplying b by a scalar c Adding multiples of A to b only 45 The inverse exists if and only if: a A has full rank but det(A)=0 b det(A) ≠ 0 c A is symmetric and det(A)=0 46 What is the purpose of the adjugate in A^{-1}? a To compute eigenvalues b To form A^{-1} = adj(A)/det(A) c To compute determinant 47 The set of all invertible matrices forms: a A group under multiplication b A vector space c A ring with zero divisors 48 Which method would you use to solve Ax=b without inverses for large systems? a Analytical inversion b Iterative methods, e.g., Gauss-Seidel c Determinant expansion 49 Which best describes a solved system by Gauss elimination? a Only determine determinant b Direct formula for x c Find values by back substitution 50 What does back substitution start from in a triangular system? a Last equation to first b Middle to edges c First to last

Matrices V Lowersixth Science Further Mathematics

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Matrices V Lowersixth Science Further MathematicsOnline version

by YAKILI LMS

What is the inverse of a 2x2 matrix [a b; c d] if ad - bc ≠ 0?

Which property holds for any invertible matrix A?

If A B = I, what is B?

The determinant of A is zero. What can be said about A^{-1}?

Which matrix operation gives the inverse of a 2x2 matrix [a b; c d]?

Cramer's rule applies to systems with how many solutions in terms of determinant?

Gaussian elimination transforms a system into which form?

What does Gauss-Jordan elimination produce for the augmented matrix?

To solve Ax = b by Gaussian elimination, you augment A with b.

Cramer's rule requires which condition on A for a unique solution?

If x_i is a solution via Cramer's rule, what is used in the numerator?

What is the property of (A^{-1})^{-1}?

The product of A and its inverse equals:

If A is 3x3 and det(A) = 2, what is det(A^{-1})?

Which method finds x in Ax=b without matrix inversion?

Which statement is true about inverse of A?

Which of the following is a necessary condition for A to be invertible?

Cramer’s rule solves for x1 in which form?

What is the effect of row operations on det(A)?

To check invertibility, you check if det(A) is:

Which is not a standard row operation?

In Ax=b, if A is 2x2, how many pivots maximum?

What is the identity matrix of size n?

What does det(A) measure in a linear system?

If A is invertible, what can be said about Ax = b?

What is a characteristic of a singular matrix?

Which matrix operation yields A^{-1} if possible?

In 2x2, if det(A) = -3, det(A^{-1}) is:

Gauss elimination aims to produce:

What is required to apply Cramer's rule for a 3x3 system?

The inverse of a matrix A is:

What is A^{-1} used to compute in Ax=b?

Which method is often used to solve Ax=b without calculating A^{-1}?

What does det(A) ≠ 0 imply about the linear transformation?

Cramer's rule expresses x_i as a ratio of:

Which is not a typical property of A^{-1}?

If A is invertible, then the system Ax=b has:

What form does Gauss elimination aim to achieve for the coefficient matrix?

What does the augmented matrix [A|b] represent?

In a 3x3 system with det(A) ≠ 0, how many pivots exist?

What is the effect of swapping two rows on det(A)?

What is the central idea of Cramer's rule?

If A is 2x2 with det(A)=4, det(A^{-1}) equals:

Which operation preserves the solution set of Ax=b?

The inverse exists if and only if:

What is the purpose of the adjugate in A^{-1}?

The set of all invertible matrices forms:

Which method would you use to solve Ax=b without inverses for large systems?

Which best describes a solved system by Gauss elimination?

What does back substitution start from in a triangular system?