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Question

The inverse of matrix A equals open square brackets table row 0 1 0 row 1 0 0 row 0 0 1 end table close square bracketsis

  1. A    
  2. A to the power of T end exponent    
  3. open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets    
  4. open square brackets table row 1 0 0 row 1 0 0 row 0 1 0 end table close square brackets    

The correct answer is: A


    A equals open square brackets table row 0 1 0 row 1 0 0 row 0 0 1 end table close square brackets rightwards double arrow vertical line A vertical line equals negative 1 left parenthesis 1 plus 0 right parenthesis equals negative 1
    a d j left parenthesis A right parenthesis equals open square brackets table row cell A subscript 11 end subscript end cell cell A subscript 21 end subscript end cell cell A subscript 31 end subscript end cell row cell A subscript 12 end subscript end cell cell A subscript 22 end subscript end cell cell A subscript 32 end subscript end cell row cell A subscript 13 end subscript end cell cell A subscript 23 end subscript end cell cell A subscript 33 end subscript end cell end table close square brackets
    fraction numerator left parenthesis negative 1 right parenthesis to the power of 3 plus 2 end exponent M subscript 32 end subscript over denominator negative 2 end fraction comma text end text equals fraction numerator negative left parenthesis negative 2 right parenthesis over denominator negative 2 end fraction equals negative 1 A subscript 11 end subscript equals 0 comma A subscript 12 end subscript equals negative 1 comma A subscript 13 end subscript equals 0
    A subscript 21 end subscript equals negative 1 comma A subscript 22 end subscript equals 0 comma A subscript 23 end subscript equals 0
    A subscript 31 end subscript equals 0, A subscript 32 end subscript equals 0 comma A subscript 33 end subscript equals negative 1
    fraction numerator left parenthesis negative 1 right parenthesis to the power of 3 plus 2 end exponent M subscript 32 end subscript over denominator negative 2 end fraction comma text end text equals fraction numerator negative left parenthesis negative 2 right parenthesis over denominator negative 2 end fraction equals negative 1 blank A to the power of negative 1 end exponent equals fraction numerator a d j left parenthesis A right parenthesis over denominator vertical line A vertical line end fraction equals open square brackets table row 0 1 0 row 1 0 0 row 0 0 1 end table close square brackets equals A.

    A real number multiplicative inverse is the number that, when multiplied by the original number, produces 1 (the identity). Because a× 1/a= 1 is the multiplicative inverse of a. When the inverse of a matrix is multiplied by a given matrix, it produces a multiplicative identity. For example, the inverse of a matrix A is A-1 and A.A-1=A-1.A=I, where I is the identity matrix.
    A square matrix with one on the diagonal and zeros everywhere else is known as an identity matrix. Think of the identity matrix as the matrix's prime number.
    An invertible matrix is one for which it is possible to calculate the inverse matrix and for which the determinant is not zero.

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